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Find the roots of the quadratic equation: 5x^2 + -3x + -2 = 0 | 1. Identify coefficients: a=5, b=-3, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -40 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √49) / 10.
5. Calculate roots: x1 = 1.0, x2 = -0.4.
Final Answer: x = 1.0, -0.4 |
Multiply the following 2x2 matrices:
A =
[-3, -3]
[4, 3]
B =
[-5, 4]
[-5, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*-5) + (-3*-5) = 30
C12 = (A11 * B12) + (A12 * B22) = (-3*4) + (-3*3) = -21
C21 = (A21 * B11) + (A22 * B21) = (4*-5) + (3*-5) = -35
C22 = (A21 * B12) + (A22 * B22) = (4*4) + (3*3) = 25
Res... |
Compute the indefinite integral of f(x) = 2x^4. | To find the indefinite integral ∫ 2x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 2 / 5 = 2/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^4 dx = 2/5x^5 + C |
Multiply the following 2x2 matrices:
A =
[0, 0]
[-1, 5]
B =
[2, 1]
[-2, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*2) + (0*-2) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*1) + (0*0) = 0
C21 = (A21 * B11) + (A22 * B21) = (-1*2) + (5*-2) = -12
C22 = (A21 * B12) + (A22 * B22) = (-1*1) + (5*0) = -1
Result Mat... |
Find the roots of the quadratic equation: 2x^2 + 1x + -3 = 0 | 1. Identify coefficients: a=2, b=1, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -24 = 25.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √25) / 4.
5. Calculate roots: x1 = 1.0, x2 = -1.5.
Final Answer: x = 1.0, -1.5 |
In a right-angled triangle, the lengths of the two legs are a = 19 and b = 3. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 19^2 + 3^2 = c^2
2. Calculate the squares: 361 + 9 = c^2
3. Sum the squares: 370 = c^2
4. Take the square root of both sides: c = √370
Final Answ... |
Find the roots of the quadratic equation: 3x^2 + -5x + 6 = 0 | 1. Identify coefficients: a=3, b=-5, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(3)(6) = 25 - 72 = -47.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √47i) / 6.
Final Answer: x = 0.8333333333333334 ± 1.1426091000668406i |
Compute the quotient and remainder of 6954 divided by 66. Show the long division steps. | To divide 6954 by 66:
1. Determine how many times 66 goes into the leading digits of 6954.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 6954 / 66 = 105 with a remainder of 24.
Proof: (105 * 66) + 2... |
Add the fractions 8/8 and 15/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 15, which is 120.
2. Convert 8/8 by multiplying top and bottom by 15: 120/120.
3. Convert 15/15 by multiplying top and bottom by 8: 120/120.
4. Add the numerators: 120 + 120 = 240.
5. The resulting fraction is 240/120.
6. S... |
Find the roots of the quadratic equation: 3x^2 + -7x + -9 = 0 | 1. Identify coefficients: a=3, b=-7, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -108 = 157.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √157) / 6.
5. Calculate roots: x1 = 3.2549940143569445, x2 = -0.9216606810236113.
Final ... |
Find the derivative of f(x) = (3x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Compute the indefinite integral of f(x) = 1x^1. | To find the indefinite integral ∫ 1x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 1 / 2 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^1 dx = 1/2x^2 + C |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 13. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 12^2 + 13^2 = c^2
2. Calculate the squares: 144 + 169 = c^2
3. Sum the squares: 313 = c^2
4. Take the square root of both sides: c = √313
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 8, 9, 10, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 8 + 9 + 10 + 10) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.2, -1.8, -0.8, 0.2, 0.2]
Squared Differences: [4.84, 3.24, 0.64, 0.04, 0.04]
Su... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 6, 1, 20, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 6 + 1 + 20 + 10) / 5 = 39 / 5 = 7.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.8, -1.8, -6.8, 12.2, 2.2]
Squared Differences: [33.64, 3.24, 46.24, 148.84, 4.84]
... |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Find the roots of the quadratic equation: 8x^2 + -5x + 5 = 0 | 1. Identify coefficients: a=8, b=-5, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(8)(5) = 25 - 160 = -135.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √135i) / 16.
Final Answer: x = 0.3125 ± 0.7261843774138906i |
In a right-angled triangle, the lengths of the two legs are a = 18 and b = 17. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 18^2 + 17^2 = c^2
2. Calculate the squares: 324 + 289 = c^2
3. Sum the squares: 613 = c^2
4. Take the square root of both sides: c = √613
Final A... |
Compute the indefinite integral of f(x) = 3x^3. | To find the indefinite integral ∫ 3x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 3 / 4 = 3/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^3 dx = 3/4x^4 + C |
Compute the quotient and remainder of 85236 divided by 23. Show the long division steps. | To divide 85236 by 23:
1. Determine how many times 23 goes into the leading digits of 85236.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 85236 / 23 = 3705 with a remainder of 21.
Proof: (3705 * 23... |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 10x^2 + 1x + -9 = 0 | 1. Identify coefficients: a=10, b=1, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -360 = 361.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √361) / 20.
5. Calculate roots: x1 = 0.9, x2 = -1.0.
Final Answer: x = 0.9, -1.0 |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Find the roots of the quadratic equation: 10x^2 + -3x + -9 = 0 | 1. Identify coefficients: a=10, b=-3, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -360 = 369.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √369) / 20.
5. Calculate roots: x1 = 1.1104686356149274, x2 = -0.8104686356149273.
Final... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 9, 2, 2, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 9 + 2 + 2 + 2) / 5 = 32 / 5 = 6.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [10.6, 2.6, -4.4, -4.4, -4.4]
Squared Differences: [112.36, 6.76, 19.36, 19.36, 19.36]
... |
Add the fractions 2/15 and 8/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 2, which is 30.
2. Convert 2/15 by multiplying top and bottom by 2: 4/30.
3. Convert 8/2 by multiplying top and bottom by 15: 120/30.
4. Add the numerators: 4 + 120 = 124.
5. The resulting fraction is 124/30.
6. Simplify b... |
Add the fractions 7/5 and 4/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 7, which is 35.
2. Convert 7/5 by multiplying top and bottom by 7: 49/35.
3. Convert 4/7 by multiplying top and bottom by 5: 20/35.
4. Add the numerators: 49 + 20 = 69.
5. The resulting fraction is 69/35.
Final Answer: 69/3... |
Evaluate sin(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 6x^2 + -9x + 5 = 0 | 1. Identify coefficients: a=6, b=-9, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = -9^2 - 4(6)(5) = 81 - 120 = -39.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (9 ± √39i) / 12.
Final Answer: x = 0.75 ± 0.5204164998665332i |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 17. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 17^2 = c^2
2. Calculate the squares: 81 + 289 = c^2
3. Sum the squares: 370 = c^2
4. Take the square root of both sides: c = √370
Final Ans... |
Find the roots of the quadratic equation: 9x^2 + -3x + -1 = 0 | 1. Identify coefficients: a=9, b=-3, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -36 = 45.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √45) / 18.
5. Calculate roots: x1 = 0.5393446629166316, x2 = -0.2060113295832983.
Final Ans... |
Add the fractions 2/15 and 9/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 7, which is 105.
2. Convert 2/15 by multiplying top and bottom by 7: 14/105.
3. Convert 9/7 by multiplying top and bottom by 15: 135/105.
4. Add the numerators: 14 + 135 = 149.
5. The resulting fraction is 149/105.
Final A... |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Add the fractions 7/6 and 11/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 13, which is 78.
2. Convert 7/6 by multiplying top and bottom by 13: 91/78.
3. Convert 11/13 by multiplying top and bottom by 6: 66/78.
4. Add the numerators: 91 + 66 = 157.
5. The resulting fraction is 157/78.
Final Answer... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 16, 15, 2, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 16 + 15 + 2 + 10) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, 4.4, 3.4, -9.6, -1.6]
Squared Differences: [11.56, 19.36, 11.56, 92.16, 2.56]... |
Find the roots of the quadratic equation: 3x^2 + 2x + -2 = 0 | 1. Identify coefficients: a=3, b=2, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -24 = 28.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √28) / 6.
5. Calculate roots: x1 = 0.5485837703548636, x2 = -1.2152504370215302.
Final Answ... |
Compute the indefinite integral of f(x) = 8x^1. | To find the indefinite integral ∫ 8x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 14, 11, 9, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 14 + 11 + 9 + 10) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.0, 2.0, -1.0, -3.0, -2.0]
Squared Differences: [16.0, 4.0, 1.0, 9.0, 4.0]
Sum... |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 61876 divided by 91. Show the long division steps. | To divide 61876 by 91:
1. Determine how many times 91 goes into the leading digits of 61876.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 61876 / 91 = 679 with a remainder of 87.
Proof: (679 * 91) ... |
Evaluate the logarithm: log_3(9) | To evaluate log_3(9), we ask the question: '3 raised to what power equals 9?'
Let x be the unknown power: 3^x = 9
Since 3^2 = 9, it follows that x = 2.
Final Answer: 2 |
Compute the indefinite integral of f(x) = 3x^5. | To find the indefinite integral ∫ 3x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 3 / 6 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^5 dx = 1/2x^6 + C |
Find the roots of the quadratic equation: 1x^2 + 2x + 8 = 0 | 1. Identify coefficients: a=1, b=2, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(1)(8) = 4 - 32 = -28.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √28i) / 2.
Final Answer: x = -1.0 ± 2.6457513110645907i |
In a right-angled triangle, the lengths of the two legs are a = 14 and b = 15. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 14^2 + 15^2 = c^2
2. Calculate the squares: 196 + 225 = c^2
3. Sum the squares: 421 = c^2
4. Take the square root of both sides: c = √421
Final A... |
Multiply the following 2x2 matrices:
A =
[5, -2]
[-5, 2]
B =
[-5, 0]
[-3, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*-5) + (-2*-3) = -19
C12 = (A11 * B12) + (A12 * B22) = (5*0) + (-2*-2) = 4
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (2*-3) = 19
C22 = (A21 * B12) + (A22 * B22) = (-5*0) + (2*-2) = -4
Res... |
Find the roots of the quadratic equation: 5x^2 + -8x + -2 = 0 | 1. Identify coefficients: a=5, b=-8, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -40 = 104.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √104) / 10.
5. Calculate roots: x1 = 1.8198039027185569, x2 = -0.2198039027185569.
Final ... |
Add the fractions 3/4 and 2/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 9, which is 36.
2. Convert 3/4 by multiplying top and bottom by 9: 27/36.
3. Convert 2/9 by multiplying top and bottom by 4: 8/36.
4. Add the numerators: 27 + 8 = 35.
5. The resulting fraction is 35/36.
Final Answer: 35/36 |
Find the roots of the quadratic equation: 9x^2 + 3x + 4 = 0 | 1. Identify coefficients: a=9, b=3, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(9)(4) = 9 - 144 = -135.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √135i) / 18.
Final Answer: x = -0.16666666666666666 ± 0.6454972243679028i |
In a right-angled triangle, the lengths of the two legs are a = 10 and b = 14. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 10^2 + 14^2 = c^2
2. Calculate the squares: 100 + 196 = c^2
3. Sum the squares: 296 = c^2
4. Take the square root of both sides: c = √296
Final A... |
Multiply the following 2x2 matrices:
A =
[3, 0]
[0, 5]
B =
[-3, 4]
[-4, 4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*-3) + (0*-4) = -9
C12 = (A11 * B12) + (A12 * B22) = (3*4) + (0*4) = 12
C21 = (A21 * B11) + (A22 * B21) = (0*-3) + (5*-4) = -20
C22 = (A21 * B12) + (A22 * B22) = (0*4) + (5*4) = 20
Result M... |
Find the roots of the quadratic equation: 2x^2 + 1x + -1 = 0 | 1. Identify coefficients: a=2, b=1, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -8 = 9.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √9) / 4.
5. Calculate roots: x1 = 0.5, x2 = -1.0.
Final Answer: x = 0.5, -1.0 |
Find the roots of the quadratic equation: 3x^2 + 3x + -4 = 0 | 1. Identify coefficients: a=3, b=3, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -48 = 57.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √57) / 6.
5. Calculate roots: x1 = 0.7583057392117917, x2 = -1.7583057392117916.
Final Answ... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
Find the roots of the quadratic equation: 10x^2 + -5x + -10 = 0 | 1. Identify coefficients: a=10, b=-5, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -400 = 425.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √425) / 20.
5. Calculate roots: x1 = 1.2807764064044151, x2 = -0.7807764064044151.
Fin... |
Multiply the following 2x2 matrices:
A =
[-1, -4]
[-4, 0]
B =
[4, -4]
[5, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*4) + (-4*5) = -24
C12 = (A11 * B12) + (A12 * B22) = (-1*-4) + (-4*0) = 4
C21 = (A21 * B11) + (A22 * B21) = (-4*4) + (0*5) = -16
C22 = (A21 * B12) + (A22 * B22) = (-4*-4) + (0*0) = 16
Resu... |
Compute the quotient and remainder of 63324 divided by 56. Show the long division steps. | To divide 63324 by 56:
1. Determine how many times 56 goes into the leading digits of 63324.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 63324 / 56 = 1130 with a remainder of 44.
Proof: (1130 * 56... |
In a right-angled triangle, the lengths of the two legs are a = 19 and b = 9. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 19^2 + 9^2 = c^2
2. Calculate the squares: 361 + 81 = c^2
3. Sum the squares: 442 = c^2
4. Take the square root of both sides: c = √442
Final Ans... |
Find the roots of the quadratic equation: 5x^2 + -3x + -4 = 0 | 1. Identify coefficients: a=5, b=-3, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -80 = 89.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √89) / 10.
5. Calculate roots: x1 = 1.2433981132056604, x2 = -0.6433981132056603.
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 7, 14, 16, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 7 + 14 + 16 + 7) / 5 = 54 / 5 = 10.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.8, -3.8, 3.2, 5.2, -3.8]
Squared Differences: [0.64, 14.44, 10.24, 27.04, 14.44]... |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Evaluate cos(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 7 and b = 18. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 7^2 + 18^2 = c^2
2. Calculate the squares: 49 + 324 = c^2
3. Sum the squares: 373 = c^2
4. Take the square root of both sides: c = √373
Final Ans... |
Find the roots of the quadratic equation: 7x^2 + 4x + -6 = 0 | 1. Identify coefficients: a=7, b=4, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -168 = 184.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √184) / 14.
5. Calculate roots: x1 = 0.683189997589324, x2 = -1.2546185690178955.
Final ... |
Multiply the following 2x2 matrices:
A =
[1, 5]
[-3, -5]
B =
[4, 1]
[-1, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*4) + (5*-1) = -1
C12 = (A11 * B12) + (A12 * B22) = (1*1) + (5*-3) = -14
C21 = (A21 * B11) + (A22 * B21) = (-3*4) + (-5*-1) = -7
C22 = (A21 * B12) + (A22 * B22) = (-3*1) + (-5*-3) = 12
Resu... |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 14, 19, 10, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 14 + 19 + 10 + 13) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.4, 1.6, 6.6, -2.4, 0.6]
Squared Differences: [40.96, 2.56, 43.56, 5.76, 0.36]
... |
Compute the indefinite integral of f(x) = 9x^3. | To find the indefinite integral ∫ 9x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 9 / 4 = 9/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^3 dx = 9/4x^4 + C |
Find the roots of the quadratic equation: 4x^2 + -4x + 3 = 0 | 1. Identify coefficients: a=4, b=-4, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(4)(3) = 16 - 48 = -32.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √32i) / 8.
Final Answer: x = 0.5 ± 0.7071067811865476i |
Compute the indefinite integral of f(x) = 5x^2. | To find the indefinite integral ∫ 5x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 5 / 3 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^2 dx = 5/3x^3 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 4, 19, 18, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 4 + 19 + 18 + 8) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.0, -8.0, 7.0, 6.0, -4.0]
Squared Differences: [1.0, 64.0, 49.0, 36.0, 16.0]
S... |
Add the fractions 12/6 and 5/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 9, which is 18.
2. Convert 12/6 by multiplying top and bottom by 3: 36/18.
3. Convert 5/9 by multiplying top and bottom by 2: 10/18.
4. Add the numerators: 36 + 10 = 46.
5. The resulting fraction is 46/18.
6. Simplify by di... |
Multiply the following 2x2 matrices:
A =
[-4, 2]
[-3, 1]
B =
[-2, -5]
[5, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-2) + (2*5) = 18
C12 = (A11 * B12) + (A12 * B22) = (-4*-5) + (2*-3) = 14
C21 = (A21 * B11) + (A22 * B21) = (-3*-2) + (1*5) = 11
C22 = (A21 * B12) + (A22 * B22) = (-3*-5) + (1*-3) = 12
Res... |
Multiply the following 2x2 matrices:
A =
[2, -2]
[4, -3]
B =
[3, -5]
[-2, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*3) + (-2*-2) = 10
C12 = (A11 * B12) + (A12 * B22) = (2*-5) + (-2*-5) = 0
C21 = (A21 * B11) + (A22 * B21) = (4*3) + (-3*-2) = 18
C22 = (A21 * B12) + (A22 * B22) = (4*-5) + (-3*-5) = -5
Resu... |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Add the fractions 12/12 and 9/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 3, which is 12.
2. Convert 12/12 by multiplying top and bottom by 1: 12/12.
3. Convert 9/3 by multiplying top and bottom by 4: 36/12.
4. Add the numerators: 12 + 36 = 48.
5. The resulting fraction is 48/12.
6. Simplify by ... |
Evaluate tan(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.7321 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 10x^2 + 6x + 8 = 0 | 1. Identify coefficients: a=10, b=6, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(10)(8) = 36 - 320 = -284.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √284i) / 20.
Final Answer: x = -0.3 ± 0.8426149773176359i |
Compute the quotient and remainder of 23653 divided by 9. Show the long division steps. | To divide 23653 by 9:
1. Determine how many times 9 goes into the leading digits of 23653.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 23653 / 9 = 2628 with a remainder of 1.
Proof: (2628 * 9) + 1... |
Add the fractions 4/14 and 10/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 14, which is 14.
2. Convert 4/14 by multiplying top and bottom by 1: 4/14.
3. Convert 10/14 by multiplying top and bottom by 1: 10/14.
4. Add the numerators: 4 + 10 = 14.
5. The resulting fraction is 14/14.
6. Simplify by ... |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Add the fractions 12/12 and 15/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 9, which is 36.
2. Convert 12/12 by multiplying top and bottom by 3: 36/36.
3. Convert 15/9 by multiplying top and bottom by 4: 60/36.
4. Add the numerators: 36 + 60 = 96.
5. The resulting fraction is 96/36.
6. Simplify by... |
In a right-angled triangle, the lengths of the two legs are a = 19 and b = 11. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 19^2 + 11^2 = c^2
2. Calculate the squares: 361 + 121 = c^2
3. Sum the squares: 482 = c^2
4. Take the square root of both sides: c = √482
Final A... |
Compute the indefinite integral of f(x) = 4x^3. | To find the indefinite integral ∫ 4x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 4 / 4 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^3 dx = 1x^4 + C |
Compute the indefinite integral of f(x) = 4x^1. | To find the indefinite integral ∫ 4x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 4 / 2 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^1 dx = 2x^2 + C |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 12. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 12^2 = c^2
2. Calculate the squares: 81 + 144 = c^2
3. Sum the squares: 225 = c^2
4. Take the square root of both sides: c = √225
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 20. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 12^2 + 20^2 = c^2
2. Calculate the squares: 144 + 400 = c^2
3. Sum the squares: 544 = c^2
4. Take the square root of both sides: c = √544
Final A... |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
Add the fractions 4/15 and 1/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 9, which is 45.
2. Convert 4/15 by multiplying top and bottom by 3: 12/45.
3. Convert 1/9 by multiplying top and bottom by 5: 5/45.
4. Add the numerators: 12 + 5 = 17.
5. The resulting fraction is 17/45.
Final Answer: 17/4... |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Find the roots of the quadratic equation: 10x^2 + -3x + -1 = 0 | 1. Identify coefficients: a=10, b=-3, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -40 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √49) / 20.
5. Calculate roots: x1 = 0.5, x2 = -0.2.
Final Answer: x = 0.5, -0.2 |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
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