ID int64 1 10k | Equation stringlengths 17 19 | Standard Form stringlengths 17 19 | Discriminant int64 -324 460 | Roots stringlengths 5 56 | Root Type stringclasses 3
values | Solution Steps stringlengths 211 265 |
|---|---|---|---|---|---|---|
8,601 | 1x^2 - 5x + 2 = 0 | 1x^2 - 5x + 2 = 0 | 17 | x = 5/2 - sqrt(17)/2, x = sqrt(17)/2 + 5/2 | Real and Distinct | Step 1: Identify coefficients a = 1, b = -5, c = 2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 17. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 5/2 - sqrt(17)/2, x = sqrt(17)/2 + 5/2. |
8,602 | 8x^2 - 2x - 7 = 0 | 8x^2 - 2x - 7 = 0 | 228 | x = 1/8 - sqrt(57)/8, x = 1/8 + sqrt(57)/8 | Real and Distinct | Step 1: Identify coefficients a = 8, b = -2, c = -7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 228. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/8 - sqrt(57)/8, x = 1/8 + sqrt(57)/8. |
8,603 | 1x^2 + 0x + 7 = 0 | 1x^2 + 0x + 7 = 0 | -28 | x = -sqrt(7)*I, x = sqrt(7)*I | Complex | Step 1: Identify coefficients a = 1, b = 0, c = 7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -28. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -sqrt(7)*I, x = sqrt(7)*I. |
8,604 | 8x^2 + 2x + 8 = 0 | 8x^2 + 2x + 8 = 0 | -252 | x = -1/8 - 3*sqrt(7)*I/8, x = -1/8 + 3*sqrt(7)*I/8 | Complex | Step 1: Identify coefficients a = 8, b = 2, c = 8. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -252. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/8 - 3*sqrt(7)*I/8, x = -1/8 + 3*sqrt(7)*I/8. |
8,605 | 9x^2 + 4x + 0 = 0 | 9x^2 + 4x + 0 = 0 | 16 | x = -4/9, x = 0 | Real and Distinct | Step 1: Identify coefficients a = 9, b = 4, c = 0. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 16. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -4/9, x = 0. |
8,606 | 7x^2 - 10x - 3 = 0 | 7x^2 - 10x - 3 = 0 | 184 | x = 5/7 - sqrt(46)/7, x = 5/7 + sqrt(46)/7 | Real and Distinct | Step 1: Identify coefficients a = 7, b = -10, c = -3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 184. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 5/7 - sqrt(46)/7, x = 5/7 + sqrt(46)/7. |
8,607 | 7x^2 - 3x + 5 = 0 | 7x^2 - 3x + 5 = 0 | -131 | x = 3/14 - sqrt(131)*I/14, x = 3/14 + sqrt(131)*I/14 | Complex | Step 1: Identify coefficients a = 7, b = -3, c = 5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -131. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 3/14 - sqrt(131)*I/14, x = 3/14 + sqrt(131)*I/14. |
8,608 | 3x^2 - 3x - 9 = 0 | 3x^2 - 3x - 9 = 0 | 117 | x = 1/2 - sqrt(13)/2, x = 1/2 + sqrt(13)/2 | Real and Distinct | Step 1: Identify coefficients a = 3, b = -3, c = -9. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 117. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/2 - sqrt(13)/2, x = 1/2 + sqrt(13)/2. |
8,609 | 1x^2 - 3x - 8 = 0 | 1x^2 - 3x - 8 = 0 | 41 | x = 3/2 - sqrt(41)/2, x = 3/2 + sqrt(41)/2 | Real and Distinct | Step 1: Identify coefficients a = 1, b = -3, c = -8. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 41. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 3/2 - sqrt(41)/2, x = 3/2 + sqrt(41)/2. |
8,610 | 3x^2 + 6x + 6 = 0 | 3x^2 + 6x + 6 = 0 | -36 | x = -1 - I, x = -1 + I | Complex | Step 1: Identify coefficients a = 3, b = 6, c = 6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -36. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1 - I, x = -1 + I. |
8,611 | 5x^2 + 4x - 4 = 0 | 5x^2 + 4x - 4 = 0 | 96 | x = -2/5 + 2*sqrt(6)/5, x = -2*sqrt(6)/5 - 2/5 | Real and Distinct | Step 1: Identify coefficients a = 5, b = 4, c = -4. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 96. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -2/5 + 2*sqrt(6)/5, x = -2*sqrt(6)/5 - 2/5. |
8,612 | 4x^2 + 6x + 9 = 0 | 4x^2 + 6x + 9 = 0 | -108 | x = -3/4 - 3*sqrt(3)*I/4, x = -3/4 + 3*sqrt(3)*I/4 | Complex | Step 1: Identify coefficients a = 4, b = 6, c = 9. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -108. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -3/4 - 3*sqrt(3)*I/4, x = -3/4 + 3*sqrt(3)*I/4. |
8,613 | 7x^2 - 9x + 4 = 0 | 7x^2 - 9x + 4 = 0 | -31 | x = 9/14 - sqrt(31)*I/14, x = 9/14 + sqrt(31)*I/14 | Complex | Step 1: Identify coefficients a = 7, b = -9, c = 4. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -31. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 9/14 - sqrt(31)*I/14, x = 9/14 + sqrt(31)*I/14. |
8,614 | 3x^2 + 4x - 6 = 0 | 3x^2 + 4x - 6 = 0 | 88 | x = -2/3 + sqrt(22)/3, x = -sqrt(22)/3 - 2/3 | Real and Distinct | Step 1: Identify coefficients a = 3, b = 4, c = -6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 88. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -2/3 + sqrt(22)/3, x = -sqrt(22)/3 - 2/3. |
8,615 | 8x^2 - 4x - 2 = 0 | 8x^2 - 4x - 2 = 0 | 80 | x = 1/4 - sqrt(5)/4, x = 1/4 + sqrt(5)/4 | Real and Distinct | Step 1: Identify coefficients a = 8, b = -4, c = -2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 80. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/4 - sqrt(5)/4, x = 1/4 + sqrt(5)/4. |
8,616 | 3x^2 + 1x + 6 = 0 | 3x^2 + 1x + 6 = 0 | -71 | x = -1/6 - sqrt(71)*I/6, x = -1/6 + sqrt(71)*I/6 | Complex | Step 1: Identify coefficients a = 3, b = 1, c = 6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -71. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/6 - sqrt(71)*I/6, x = -1/6 + sqrt(71)*I/6. |
8,617 | 5x^2 + 6x - 1 = 0 | 5x^2 + 6x - 1 = 0 | 56 | x = -3/5 + sqrt(14)/5, x = -sqrt(14)/5 - 3/5 | Real and Distinct | Step 1: Identify coefficients a = 5, b = 6, c = -1. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 56. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -3/5 + sqrt(14)/5, x = -sqrt(14)/5 - 3/5. |
8,618 | 2x^2 + 5x - 10 = 0 | 2x^2 + 5x - 10 = 0 | 105 | x = -5/4 + sqrt(105)/4, x = -sqrt(105)/4 - 5/4 | Real and Distinct | Step 1: Identify coefficients a = 2, b = 5, c = -10. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 105. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -5/4 + sqrt(105)/4, x = -sqrt(105)/4 - 5/4. |
8,619 | 8x^2 - 1x + 5 = 0 | 8x^2 - 1x + 5 = 0 | -159 | x = 1/16 - sqrt(159)*I/16, x = 1/16 + sqrt(159)*I/16 | Complex | Step 1: Identify coefficients a = 8, b = -1, c = 5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -159. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/16 - sqrt(159)*I/16, x = 1/16 + sqrt(159)*I/16. |
8,620 | 5x^2 + 5x - 9 = 0 | 5x^2 + 5x - 9 = 0 | 205 | x = -1/2 + sqrt(205)/10, x = -sqrt(205)/10 - 1/2 | Real and Distinct | Step 1: Identify coefficients a = 5, b = 5, c = -9. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 205. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/2 + sqrt(205)/10, x = -sqrt(205)/10 - 1/2. |
8,621 | 2x^2 + 7x + 3 = 0 | 2x^2 + 7x + 3 = 0 | 25 | x = -3, x = -1/2 | Real and Distinct | Step 1: Identify coefficients a = 2, b = 7, c = 3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 25. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -3, x = -1/2. |
8,622 | 1x^2 - 7x + 8 = 0 | 1x^2 - 7x + 8 = 0 | 17 | x = 7/2 - sqrt(17)/2, x = sqrt(17)/2 + 7/2 | Real and Distinct | Step 1: Identify coefficients a = 1, b = -7, c = 8. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 17. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 7/2 - sqrt(17)/2, x = sqrt(17)/2 + 7/2. |
8,623 | 2x^2 + 4x + 5 = 0 | 2x^2 + 4x + 5 = 0 | -24 | x = -1 - sqrt(6)*I/2, x = -1 + sqrt(6)*I/2 | Complex | Step 1: Identify coefficients a = 2, b = 4, c = 5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -24. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1 - sqrt(6)*I/2, x = -1 + sqrt(6)*I/2. |
8,624 | 3x^2 + 9x - 10 = 0 | 3x^2 + 9x - 10 = 0 | 201 | x = -3/2 + sqrt(201)/6, x = -sqrt(201)/6 - 3/2 | Real and Distinct | Step 1: Identify coefficients a = 3, b = 9, c = -10. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 201. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -3/2 + sqrt(201)/6, x = -sqrt(201)/6 - 3/2. |
8,625 | 9x^2 - 9x + 3 = 0 | 9x^2 - 9x + 3 = 0 | -27 | x = 1/2 - sqrt(3)*I/6, x = 1/2 + sqrt(3)*I/6 | Complex | Step 1: Identify coefficients a = 9, b = -9, c = 3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -27. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/2 - sqrt(3)*I/6, x = 1/2 + sqrt(3)*I/6. |
8,626 | 7x^2 - 4x + 9 = 0 | 7x^2 - 4x + 9 = 0 | -236 | x = 2/7 - sqrt(59)*I/7, x = 2/7 + sqrt(59)*I/7 | Complex | Step 1: Identify coefficients a = 7, b = -4, c = 9. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -236. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 2/7 - sqrt(59)*I/7, x = 2/7 + sqrt(59)*I/7. |
8,627 | 3x^2 - 6x - 10 = 0 | 3x^2 - 6x - 10 = 0 | 156 | x = 1 - sqrt(39)/3, x = 1 + sqrt(39)/3 | Real and Distinct | Step 1: Identify coefficients a = 3, b = -6, c = -10. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 156. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1 - sqrt(39)/3, x = 1 + sqrt(39)/3. |
8,628 | 7x^2 - 5x + 8 = 0 | 7x^2 - 5x + 8 = 0 | -199 | x = 5/14 - sqrt(199)*I/14, x = 5/14 + sqrt(199)*I/14 | Complex | Step 1: Identify coefficients a = 7, b = -5, c = 8. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -199. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 5/14 - sqrt(199)*I/14, x = 5/14 + sqrt(199)*I/14. |
8,629 | 8x^2 - 6x + 3 = 0 | 8x^2 - 6x + 3 = 0 | -60 | x = 3/8 - sqrt(15)*I/8, x = 3/8 + sqrt(15)*I/8 | Complex | Step 1: Identify coefficients a = 8, b = -6, c = 3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -60. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 3/8 - sqrt(15)*I/8, x = 3/8 + sqrt(15)*I/8. |
8,630 | 1x^2 + 7x + 0 = 0 | 1x^2 + 7x + 0 = 0 | 49 | x = -7, x = 0 | Real and Distinct | Step 1: Identify coefficients a = 1, b = 7, c = 0. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 49. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -7, x = 0. |
8,631 | 3x^2 - 2x + 5 = 0 | 3x^2 - 2x + 5 = 0 | -56 | x = 1/3 - sqrt(14)*I/3, x = 1/3 + sqrt(14)*I/3 | Complex | Step 1: Identify coefficients a = 3, b = -2, c = 5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -56. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/3 - sqrt(14)*I/3, x = 1/3 + sqrt(14)*I/3. |
8,632 | 1x^2 - 10x + 4 = 0 | 1x^2 - 10x + 4 = 0 | 84 | x = 5 - sqrt(21), x = sqrt(21) + 5 | Real and Distinct | Step 1: Identify coefficients a = 1, b = -10, c = 4. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 84. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 5 - sqrt(21), x = sqrt(21) + 5. |
8,633 | 4x^2 - 4x - 6 = 0 | 4x^2 - 4x - 6 = 0 | 112 | x = 1/2 - sqrt(7)/2, x = 1/2 + sqrt(7)/2 | Real and Distinct | Step 1: Identify coefficients a = 4, b = -4, c = -6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 112. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/2 - sqrt(7)/2, x = 1/2 + sqrt(7)/2. |
8,634 | 4x^2 - 7x - 6 = 0 | 4x^2 - 7x - 6 = 0 | 145 | x = 7/8 - sqrt(145)/8, x = 7/8 + sqrt(145)/8 | Real and Distinct | Step 1: Identify coefficients a = 4, b = -7, c = -6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 145. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 7/8 - sqrt(145)/8, x = 7/8 + sqrt(145)/8. |
8,635 | 1x^2 - 6x + 5 = 0 | 1x^2 - 6x + 5 = 0 | 16 | x = 1, x = 5 | Real and Distinct | Step 1: Identify coefficients a = 1, b = -6, c = 5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 16. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1, x = 5. |
8,636 | 1x^2 + 7x + 1 = 0 | 1x^2 + 7x + 1 = 0 | 45 | x = -7/2 - 3*sqrt(5)/2, x = -7/2 + 3*sqrt(5)/2 | Real and Distinct | Step 1: Identify coefficients a = 1, b = 7, c = 1. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 45. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -7/2 - 3*sqrt(5)/2, x = -7/2 + 3*sqrt(5)/2. |
8,637 | 9x^2 + 8x + 4 = 0 | 9x^2 + 8x + 4 = 0 | -80 | x = -4/9 - 2*sqrt(5)*I/9, x = -4/9 + 2*sqrt(5)*I/9 | Complex | Step 1: Identify coefficients a = 9, b = 8, c = 4. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -80. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -4/9 - 2*sqrt(5)*I/9, x = -4/9 + 2*sqrt(5)*I/9. |
8,638 | 6x^2 + 3x + 7 = 0 | 6x^2 + 3x + 7 = 0 | -159 | x = -1/4 - sqrt(159)*I/12, x = -1/4 + sqrt(159)*I/12 | Complex | Step 1: Identify coefficients a = 6, b = 3, c = 7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -159. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/4 - sqrt(159)*I/12, x = -1/4 + sqrt(159)*I/12. |
8,639 | 2x^2 + 2x + 4 = 0 | 2x^2 + 2x + 4 = 0 | -28 | x = -1/2 - sqrt(7)*I/2, x = -1/2 + sqrt(7)*I/2 | Complex | Step 1: Identify coefficients a = 2, b = 2, c = 4. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -28. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/2 - sqrt(7)*I/2, x = -1/2 + sqrt(7)*I/2. |
8,640 | 1x^2 - 4x + 3 = 0 | 1x^2 - 4x + 3 = 0 | 4 | x = 1, x = 3 | Real and Distinct | Step 1: Identify coefficients a = 1, b = -4, c = 3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 4. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1, x = 3. |
8,641 | 7x^2 - 7x - 7 = 0 | 7x^2 - 7x - 7 = 0 | 245 | x = 1/2 - sqrt(5)/2, x = 1/2 + sqrt(5)/2 | Real and Distinct | Step 1: Identify coefficients a = 7, b = -7, c = -7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 245. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/2 - sqrt(5)/2, x = 1/2 + sqrt(5)/2. |
8,642 | 4x^2 - 5x - 10 = 0 | 4x^2 - 5x - 10 = 0 | 185 | x = 5/8 - sqrt(185)/8, x = 5/8 + sqrt(185)/8 | Real and Distinct | Step 1: Identify coefficients a = 4, b = -5, c = -10. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 185. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 5/8 - sqrt(185)/8, x = 5/8 + sqrt(185)/8. |
8,643 | 4x^2 - 5x + 5 = 0 | 4x^2 - 5x + 5 = 0 | -55 | x = 5/8 - sqrt(55)*I/8, x = 5/8 + sqrt(55)*I/8 | Complex | Step 1: Identify coefficients a = 4, b = -5, c = 5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -55. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 5/8 - sqrt(55)*I/8, x = 5/8 + sqrt(55)*I/8. |
8,644 | 3x^2 + 8x + 6 = 0 | 3x^2 + 8x + 6 = 0 | -8 | x = -4/3 - sqrt(2)*I/3, x = -4/3 + sqrt(2)*I/3 | Complex | Step 1: Identify coefficients a = 3, b = 8, c = 6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -8. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -4/3 - sqrt(2)*I/3, x = -4/3 + sqrt(2)*I/3. |
8,645 | 5x^2 + 1x - 5 = 0 | 5x^2 + 1x - 5 = 0 | 101 | x = -1/10 + sqrt(101)/10, x = -sqrt(101)/10 - 1/10 | Real and Distinct | Step 1: Identify coefficients a = 5, b = 1, c = -5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 101. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/10 + sqrt(101)/10, x = -sqrt(101)/10 - 1/10. |
8,646 | 9x^2 + 5x + 8 = 0 | 9x^2 + 5x + 8 = 0 | -263 | x = -5/18 - sqrt(263)*I/18, x = -5/18 + sqrt(263)*I/18 | Complex | Step 1: Identify coefficients a = 9, b = 5, c = 8. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -263. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -5/18 - sqrt(263)*I/18, x = -5/18 + sqrt(263)*I/18. |
8,647 | 1x^2 - 2x + 2 = 0 | 1x^2 - 2x + 2 = 0 | -4 | x = 1 - I, x = 1 + I | Complex | Step 1: Identify coefficients a = 1, b = -2, c = 2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -4. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1 - I, x = 1 + I. |
8,648 | 7x^2 - 6x + 2 = 0 | 7x^2 - 6x + 2 = 0 | -20 | x = 3/7 - sqrt(5)*I/7, x = 3/7 + sqrt(5)*I/7 | Complex | Step 1: Identify coefficients a = 7, b = -6, c = 2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -20. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 3/7 - sqrt(5)*I/7, x = 3/7 + sqrt(5)*I/7. |
8,649 | 5x^2 - 8x - 5 = 0 | 5x^2 - 8x - 5 = 0 | 164 | x = 4/5 - sqrt(41)/5, x = 4/5 + sqrt(41)/5 | Real and Distinct | Step 1: Identify coefficients a = 5, b = -8, c = -5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 164. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 4/5 - sqrt(41)/5, x = 4/5 + sqrt(41)/5. |
8,650 | 1x^2 - 5x - 5 = 0 | 1x^2 - 5x - 5 = 0 | 45 | x = 5/2 - 3*sqrt(5)/2, x = 5/2 + 3*sqrt(5)/2 | Real and Distinct | Step 1: Identify coefficients a = 1, b = -5, c = -5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 45. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 5/2 - 3*sqrt(5)/2, x = 5/2 + 3*sqrt(5)/2. |
8,651 | 3x^2 + 3x + 5 = 0 | 3x^2 + 3x + 5 = 0 | -51 | x = -1/2 - sqrt(51)*I/6, x = -1/2 + sqrt(51)*I/6 | Complex | Step 1: Identify coefficients a = 3, b = 3, c = 5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -51. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/2 - sqrt(51)*I/6, x = -1/2 + sqrt(51)*I/6. |
8,652 | 4x^2 - 10x + 8 = 0 | 4x^2 - 10x + 8 = 0 | -28 | x = 5/4 - sqrt(7)*I/4, x = 5/4 + sqrt(7)*I/4 | Complex | Step 1: Identify coefficients a = 4, b = -10, c = 8. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -28. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 5/4 - sqrt(7)*I/4, x = 5/4 + sqrt(7)*I/4. |
8,653 | 6x^2 + 5x - 10 = 0 | 6x^2 + 5x - 10 = 0 | 265 | x = -5/12 + sqrt(265)/12, x = -sqrt(265)/12 - 5/12 | Real and Distinct | Step 1: Identify coefficients a = 6, b = 5, c = -10. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 265. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -5/12 + sqrt(265)/12, x = -sqrt(265)/12 - 5/12. |
8,654 | 3x^2 + 0x - 3 = 0 | 3x^2 + 0x - 3 = 0 | 36 | x = -1, x = 1 | Real and Distinct | Step 1: Identify coefficients a = 3, b = 0, c = -3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 36. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1, x = 1. |
8,655 | 8x^2 + 9x - 3 = 0 | 8x^2 + 9x - 3 = 0 | 177 | x = -9/16 + sqrt(177)/16, x = -sqrt(177)/16 - 9/16 | Real and Distinct | Step 1: Identify coefficients a = 8, b = 9, c = -3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 177. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -9/16 + sqrt(177)/16, x = -sqrt(177)/16 - 9/16. |
8,656 | 6x^2 + 3x - 3 = 0 | 6x^2 + 3x - 3 = 0 | 81 | x = -1, x = 1/2 | Real and Distinct | Step 1: Identify coefficients a = 6, b = 3, c = -3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 81. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1, x = 1/2. |
8,657 | 7x^2 + 7x - 10 = 0 | 7x^2 + 7x - 10 = 0 | 329 | x = -1/2 + sqrt(329)/14, x = -sqrt(329)/14 - 1/2 | Real and Distinct | Step 1: Identify coefficients a = 7, b = 7, c = -10. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 329. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/2 + sqrt(329)/14, x = -sqrt(329)/14 - 1/2. |
8,658 | 5x^2 - 1x + 5 = 0 | 5x^2 - 1x + 5 = 0 | -99 | x = 1/10 - 3*sqrt(11)*I/10, x = 1/10 + 3*sqrt(11)*I/10 | Complex | Step 1: Identify coefficients a = 5, b = -1, c = 5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -99. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/10 - 3*sqrt(11)*I/10, x = 1/10 + 3*sqrt(11)*I/10. |
8,659 | 2x^2 - 3x - 5 = 0 | 2x^2 - 3x - 5 = 0 | 49 | x = -1, x = 5/2 | Real and Distinct | Step 1: Identify coefficients a = 2, b = -3, c = -5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 49. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1, x = 5/2. |
8,660 | 8x^2 - 8x - 5 = 0 | 8x^2 - 8x - 5 = 0 | 224 | x = 1/2 - sqrt(14)/4, x = 1/2 + sqrt(14)/4 | Real and Distinct | Step 1: Identify coefficients a = 8, b = -8, c = -5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 224. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/2 - sqrt(14)/4, x = 1/2 + sqrt(14)/4. |
8,661 | 4x^2 + 0x - 2 = 0 | 4x^2 + 0x - 2 = 0 | 32 | x = -sqrt(2)/2, x = sqrt(2)/2 | Real and Distinct | Step 1: Identify coefficients a = 4, b = 0, c = -2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 32. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -sqrt(2)/2, x = sqrt(2)/2. |
8,662 | 3x^2 + 3x - 7 = 0 | 3x^2 + 3x - 7 = 0 | 93 | x = -1/2 + sqrt(93)/6, x = -sqrt(93)/6 - 1/2 | Real and Distinct | Step 1: Identify coefficients a = 3, b = 3, c = -7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 93. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/2 + sqrt(93)/6, x = -sqrt(93)/6 - 1/2. |
8,663 | 3x^2 + 5x + 7 = 0 | 3x^2 + 5x + 7 = 0 | -59 | x = -5/6 - sqrt(59)*I/6, x = -5/6 + sqrt(59)*I/6 | Complex | Step 1: Identify coefficients a = 3, b = 5, c = 7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -59. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -5/6 - sqrt(59)*I/6, x = -5/6 + sqrt(59)*I/6. |
8,664 | 2x^2 + 5x - 7 = 0 | 2x^2 + 5x - 7 = 0 | 81 | x = -7/2, x = 1 | Real and Distinct | Step 1: Identify coefficients a = 2, b = 5, c = -7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 81. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -7/2, x = 1. |
8,665 | 8x^2 + 6x + 6 = 0 | 8x^2 + 6x + 6 = 0 | -156 | x = -3/8 - sqrt(39)*I/8, x = -3/8 + sqrt(39)*I/8 | Complex | Step 1: Identify coefficients a = 8, b = 6, c = 6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -156. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -3/8 - sqrt(39)*I/8, x = -3/8 + sqrt(39)*I/8. |
8,666 | 8x^2 - 8x - 8 = 0 | 8x^2 - 8x - 8 = 0 | 320 | x = 1/2 - sqrt(5)/2, x = 1/2 + sqrt(5)/2 | Real and Distinct | Step 1: Identify coefficients a = 8, b = -8, c = -8. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 320. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/2 - sqrt(5)/2, x = 1/2 + sqrt(5)/2. |
8,667 | 3x^2 - 8x + 8 = 0 | 3x^2 - 8x + 8 = 0 | -32 | x = 4/3 - 2*sqrt(2)*I/3, x = 4/3 + 2*sqrt(2)*I/3 | Complex | Step 1: Identify coefficients a = 3, b = -8, c = 8. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -32. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 4/3 - 2*sqrt(2)*I/3, x = 4/3 + 2*sqrt(2)*I/3. |
8,668 | 7x^2 + 8x - 6 = 0 | 7x^2 + 8x - 6 = 0 | 232 | x = -4/7 + sqrt(58)/7, x = -sqrt(58)/7 - 4/7 | Real and Distinct | Step 1: Identify coefficients a = 7, b = 8, c = -6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 232. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -4/7 + sqrt(58)/7, x = -sqrt(58)/7 - 4/7. |
8,669 | 9x^2 + 2x - 5 = 0 | 9x^2 + 2x - 5 = 0 | 184 | x = -1/9 + sqrt(46)/9, x = -sqrt(46)/9 - 1/9 | Real and Distinct | Step 1: Identify coefficients a = 9, b = 2, c = -5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 184. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/9 + sqrt(46)/9, x = -sqrt(46)/9 - 1/9. |
8,670 | 8x^2 + 8x - 7 = 0 | 8x^2 + 8x - 7 = 0 | 288 | x = -1/2 + 3*sqrt(2)/4, x = -3*sqrt(2)/4 - 1/2 | Real and Distinct | Step 1: Identify coefficients a = 8, b = 8, c = -7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 288. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/2 + 3*sqrt(2)/4, x = -3*sqrt(2)/4 - 1/2. |
8,671 | 4x^2 + 6x + 6 = 0 | 4x^2 + 6x + 6 = 0 | -60 | x = -3/4 - sqrt(15)*I/4, x = -3/4 + sqrt(15)*I/4 | Complex | Step 1: Identify coefficients a = 4, b = 6, c = 6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -60. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -3/4 - sqrt(15)*I/4, x = -3/4 + sqrt(15)*I/4. |
8,672 | 9x^2 + 9x + 1 = 0 | 9x^2 + 9x + 1 = 0 | 45 | x = -1/2 - sqrt(5)/6, x = -1/2 + sqrt(5)/6 | Real and Distinct | Step 1: Identify coefficients a = 9, b = 9, c = 1. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 45. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/2 - sqrt(5)/6, x = -1/2 + sqrt(5)/6. |
8,673 | 1x^2 - 8x - 3 = 0 | 1x^2 - 8x - 3 = 0 | 76 | x = 4 - sqrt(19), x = 4 + sqrt(19) | Real and Distinct | Step 1: Identify coefficients a = 1, b = -8, c = -3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 76. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 4 - sqrt(19), x = 4 + sqrt(19). |
8,674 | 1x^2 - 3x + 6 = 0 | 1x^2 - 3x + 6 = 0 | -15 | x = 3/2 - sqrt(15)*I/2, x = 3/2 + sqrt(15)*I/2 | Complex | Step 1: Identify coefficients a = 1, b = -3, c = 6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -15. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 3/2 - sqrt(15)*I/2, x = 3/2 + sqrt(15)*I/2. |
8,675 | 7x^2 + 4x + 5 = 0 | 7x^2 + 4x + 5 = 0 | -124 | x = -2/7 - sqrt(31)*I/7, x = -2/7 + sqrt(31)*I/7 | Complex | Step 1: Identify coefficients a = 7, b = 4, c = 5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -124. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -2/7 - sqrt(31)*I/7, x = -2/7 + sqrt(31)*I/7. |
8,676 | 9x^2 - 3x - 2 = 0 | 9x^2 - 3x - 2 = 0 | 81 | x = -1/3, x = 2/3 | Real and Distinct | Step 1: Identify coefficients a = 9, b = -3, c = -2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 81. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/3, x = 2/3. |
8,677 | 5x^2 - 6x + 1 = 0 | 5x^2 - 6x + 1 = 0 | 16 | x = 1/5, x = 1 | Real and Distinct | Step 1: Identify coefficients a = 5, b = -6, c = 1. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 16. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/5, x = 1. |
8,678 | 7x^2 - 7x - 8 = 0 | 7x^2 - 7x - 8 = 0 | 273 | x = 1/2 - sqrt(273)/14, x = 1/2 + sqrt(273)/14 | Real and Distinct | Step 1: Identify coefficients a = 7, b = -7, c = -8. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 273. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/2 - sqrt(273)/14, x = 1/2 + sqrt(273)/14. |
8,679 | 2x^2 - 5x - 3 = 0 | 2x^2 - 5x - 3 = 0 | 49 | x = -1/2, x = 3 | Real and Distinct | Step 1: Identify coefficients a = 2, b = -5, c = -3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 49. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/2, x = 3. |
8,680 | 5x^2 - 10x - 7 = 0 | 5x^2 - 10x - 7 = 0 | 240 | x = 1 - 2*sqrt(15)/5, x = 1 + 2*sqrt(15)/5 | Real and Distinct | Step 1: Identify coefficients a = 5, b = -10, c = -7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 240. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1 - 2*sqrt(15)/5, x = 1 + 2*sqrt(15)/5. |
8,681 | 5x^2 - 10x - 1 = 0 | 5x^2 - 10x - 1 = 0 | 120 | x = 1 - sqrt(30)/5, x = 1 + sqrt(30)/5 | Real and Distinct | Step 1: Identify coefficients a = 5, b = -10, c = -1. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 120. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1 - sqrt(30)/5, x = 1 + sqrt(30)/5. |
8,682 | 4x^2 - 6x - 7 = 0 | 4x^2 - 6x - 7 = 0 | 148 | x = 3/4 - sqrt(37)/4, x = 3/4 + sqrt(37)/4 | Real and Distinct | Step 1: Identify coefficients a = 4, b = -6, c = -7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 148. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 3/4 - sqrt(37)/4, x = 3/4 + sqrt(37)/4. |
8,683 | 1x^2 + 5x + 4 = 0 | 1x^2 + 5x + 4 = 0 | 9 | x = -4, x = -1 | Real and Distinct | Step 1: Identify coefficients a = 1, b = 5, c = 4. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 9. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -4, x = -1. |
8,684 | 8x^2 + 4x - 9 = 0 | 8x^2 + 4x - 9 = 0 | 304 | x = -1/4 + sqrt(19)/4, x = -sqrt(19)/4 - 1/4 | Real and Distinct | Step 1: Identify coefficients a = 8, b = 4, c = -9. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 304. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/4 + sqrt(19)/4, x = -sqrt(19)/4 - 1/4. |
8,685 | 3x^2 - 7x + 2 = 0 | 3x^2 - 7x + 2 = 0 | 25 | x = 1/3, x = 2 | Real and Distinct | Step 1: Identify coefficients a = 3, b = -7, c = 2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 25. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/3, x = 2. |
8,686 | 2x^2 + 8x - 6 = 0 | 2x^2 + 8x - 6 = 0 | 112 | x = -2 + sqrt(7), x = -sqrt(7) - 2 | Real and Distinct | Step 1: Identify coefficients a = 2, b = 8, c = -6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 112. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -2 + sqrt(7), x = -sqrt(7) - 2. |
8,687 | 8x^2 - 2x - 6 = 0 | 8x^2 - 2x - 6 = 0 | 196 | x = -3/4, x = 1 | Real and Distinct | Step 1: Identify coefficients a = 8, b = -2, c = -6. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 196. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -3/4, x = 1. |
8,688 | 6x^2 + 3x - 2 = 0 | 6x^2 + 3x - 2 = 0 | 57 | x = -1/4 + sqrt(57)/12, x = -sqrt(57)/12 - 1/4 | Real and Distinct | Step 1: Identify coefficients a = 6, b = 3, c = -2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 57. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/4 + sqrt(57)/12, x = -sqrt(57)/12 - 1/4. |
8,689 | 5x^2 - 3x - 3 = 0 | 5x^2 - 3x - 3 = 0 | 69 | x = 3/10 - sqrt(69)/10, x = 3/10 + sqrt(69)/10 | Real and Distinct | Step 1: Identify coefficients a = 5, b = -3, c = -3. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 69. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 3/10 - sqrt(69)/10, x = 3/10 + sqrt(69)/10. |
8,690 | 7x^2 - 4x - 2 = 0 | 7x^2 - 4x - 2 = 0 | 72 | x = 2/7 - 3*sqrt(2)/7, x = 2/7 + 3*sqrt(2)/7 | Real and Distinct | Step 1: Identify coefficients a = 7, b = -4, c = -2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 72. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 2/7 - 3*sqrt(2)/7, x = 2/7 + 3*sqrt(2)/7. |
8,691 | 8x^2 - 2x - 9 = 0 | 8x^2 - 2x - 9 = 0 | 292 | x = 1/8 - sqrt(73)/8, x = 1/8 + sqrt(73)/8 | Real and Distinct | Step 1: Identify coefficients a = 8, b = -2, c = -9. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 292. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/8 - sqrt(73)/8, x = 1/8 + sqrt(73)/8. |
8,692 | 9x^2 - 2x - 8 = 0 | 9x^2 - 2x - 8 = 0 | 292 | x = 1/9 - sqrt(73)/9, x = 1/9 + sqrt(73)/9 | Real and Distinct | Step 1: Identify coefficients a = 9, b = -2, c = -8. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 292. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/9 - sqrt(73)/9, x = 1/9 + sqrt(73)/9. |
8,693 | 8x^2 - 1x - 2 = 0 | 8x^2 - 1x - 2 = 0 | 65 | x = 1/16 - sqrt(65)/16, x = 1/16 + sqrt(65)/16 | Real and Distinct | Step 1: Identify coefficients a = 8, b = -1, c = -2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 65. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/16 - sqrt(65)/16, x = 1/16 + sqrt(65)/16. |
8,694 | 3x^2 + 9x - 1 = 0 | 3x^2 + 9x - 1 = 0 | 93 | x = -3/2 + sqrt(93)/6, x = -sqrt(93)/6 - 3/2 | Real and Distinct | Step 1: Identify coefficients a = 3, b = 9, c = -1. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 93. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -3/2 + sqrt(93)/6, x = -sqrt(93)/6 - 3/2. |
8,695 | 3x^2 - 2x + 7 = 0 | 3x^2 - 2x + 7 = 0 | -80 | x = 1/3 - 2*sqrt(5)*I/3, x = 1/3 + 2*sqrt(5)*I/3 | Complex | Step 1: Identify coefficients a = 3, b = -2, c = 7. Step 2: Calculate the discriminant Δ = b^2 - 4ac = -80. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 1/3 - 2*sqrt(5)*I/3, x = 1/3 + 2*sqrt(5)*I/3. |
8,696 | 2x^2 - 2x - 4 = 0 | 2x^2 - 2x - 4 = 0 | 36 | x = -1, x = 2 | Real and Distinct | Step 1: Identify coefficients a = 2, b = -2, c = -4. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 36. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1, x = 2. |
8,697 | 7x^2 + 9x - 5 = 0 | 7x^2 + 9x - 5 = 0 | 221 | x = -9/14 + sqrt(221)/14, x = -sqrt(221)/14 - 9/14 | Real and Distinct | Step 1: Identify coefficients a = 7, b = 9, c = -5. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 221. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -9/14 + sqrt(221)/14, x = -sqrt(221)/14 - 9/14. |
8,698 | 3x^2 - 10x + 2 = 0 | 3x^2 - 10x + 2 = 0 | 76 | x = 5/3 - sqrt(19)/3, x = sqrt(19)/3 + 5/3 | Real and Distinct | Step 1: Identify coefficients a = 3, b = -10, c = 2. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 76. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = 5/3 - sqrt(19)/3, x = sqrt(19)/3 + 5/3. |
8,699 | 8x^2 + 2x - 9 = 0 | 8x^2 + 2x - 9 = 0 | 292 | x = -1/8 + sqrt(73)/8, x = -sqrt(73)/8 - 1/8 | Real and Distinct | Step 1: Identify coefficients a = 8, b = 2, c = -9. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 292. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -1/8 + sqrt(73)/8, x = -sqrt(73)/8 - 1/8. |
8,700 | 7x^2 + 3x - 9 = 0 | 7x^2 + 3x - 9 = 0 | 261 | x = -3/14 + 3*sqrt(29)/14, x = -3*sqrt(29)/14 - 3/14 | Real and Distinct | Step 1: Identify coefficients a = 7, b = 3, c = -9. Step 2: Calculate the discriminant Δ = b^2 - 4ac = 261. Step 3: Solve the equation using the quadratic formula x = (-b ± √Δ) / (2a). Step 4: Solutions are: x = -3/14 + 3*sqrt(29)/14, x = -3*sqrt(29)/14 - 3/14. |
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