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Step 4.Write the equation that relatesxandy. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Solve a rational inequality.Step 1.Write the inequality as one quotient on the left and zero on the right.Step 2.Determine the critical pointsâthe points where the rational expression will be zero or undefined.Step 3.Use the critical points to divide the number line into intervals.Step 4.Test a value in each interval... | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Write the inequality as one quotient on the left and zero on the right. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Determine the critical pointsâthe points where the rational expression will be zero or undefined. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.Use the critical points to divide the number line into intervals. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 4.Test a value in each interval. Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 5.Determine the intervals where the inequality is correct. Write the solution in interval notation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
complex rational expression : A complex rational expression is a rational expression in which the numerator and/or denominator contains a rational expression. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-terms |
critical point of a rational inequality : The critical point of a rational inequality is a number which makes the rational expression zero or undefined. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-terms |
extraneous solution to a rational equation : An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-terms |
proportion : When two rational expressions are equal, the equation relating them is called a proportion. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-terms |
rational equation : A rational equation is an equation that contains a rational expression. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-terms |
rational expression : A rational expression is an expression of the formpq,pq,wherepandqare polynomials andqâ0.qâ0. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-terms |
rational function : A rational function is a function of the formR(x)=p(x)q(x)R(x)=p(x)q(x)wherep(x)p(x)andq(x)q(x)are polynomial functions andq(x)q(x)is not zero. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-terms |
rational inequality : A rational inequality is an inequality that contains a rational expression. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-terms |
similar figures : Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides have the same ratio. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-terms |
simplified rational expression : A simplified rational expression has no common factors, other than 1, in its numerator and denominator. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-terms |
a + b i a + b i | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-concepts |
b = 0 b = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-concepts |
a + 0 · i a a + 0 · i a | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-concepts |
a + b i a + b i | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-concepts |
a = 0 a = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-concepts |
0 + b i b i 0 + b i b i | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-concepts |
complex conjugate pair : A complex conjugate pair is of the forma+bi,aâbi. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
complex number : A complex number is of the forma+bi, whereaandbare real numbers. We callathe real part andbthe imaginary part. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
complex number system : The complex number system is made up of both the real numbers and the imaginary numbers. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
imaginary unit : The imaginary unitiiis the number whose square is â1.i2= â1 ori=â1.i=â1. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
like radicals : Like radicals are radical expressions with the same index and the same radicand. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
radical equation : An equation in which a variable is in the radicand of a radical expression is called a radical equation. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
radical function : A radical function is a function that is defined by a radical expression. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
rationalizing the denominator : Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
square of a number : Ifn2=m, thenmis the square ofn. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
square root of a number : Ifn2=m,thennis a square root ofm. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
standard form : A complex number is in standard form when written asa+bi,a+bi,wherea, bare real numbers. | https://openstax.org/books/intermediate-algebra-2e/pages/8-key-terms |
Square Root PropertyIfx2=kx2=k, thenx=korx=âkx=korx=âkorx=±kx=±kHow to solve a quadratic equation using the square root property.Step 1.Isolate the quadratic term and make its coefficient one.Step 2.Use Square Root Property.Step 3.Simplify the radical.Step 4.Check the solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Ifx2=kx2=k, thenx=korx=âkx=korx=âkorx=±kx=±k | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Isolate the quadratic term and make its coefficient one. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Use Square Root Property. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Simplify the radical. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 4.Check the solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Binomial Squares PatternIfaandbare real numbers, | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
How to Complete a SquareStep 1.Identifyb, the coefficient ofx.Step 2.Find(12b)2,(12b)2,the number to complete the square.Step 3.Add the(12b)2(12b)2tox2+bxStep 4.Rewrite the trinomial as a binomial square | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Identifyb, the coefficient ofx. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Find(12b)2,(12b)2,the number to complete the square. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Add the(12b)2(12b)2tox2+bx | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 4.Rewrite the trinomial as a binomial square | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
How to solve a quadratic equation of the formax2+bx+c= 0 by completing the square.Step 1.Divide byato make the coefficient ofx2term 1.Step 2.Isolate the variable terms on one side and the constant terms on the other.Step 3.Find(12·b)2,(12·b)2,the number needed to complete the square. Add it to both sides of the equat... | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Divide byato make the coefficient ofx2term 1. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Isolate the variable terms on one side and the constant terms on the other. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Find(12·b)2,(12·b)2,the number needed to complete the square. Add it to both sides of the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 4.Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 5.Use the Square Root Property. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 6.Simplify the radical and then solve the two resulting equations. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 7.Check the solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Quadratic FormulaThe solutions to a quadratic equation of the formax2+bx+c= 0,aâ0aâ0are given by the formula:x=âb±b2â4ac2ax=âb±b2â4ac2a | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
The solutions to a quadratic equation of the formax2+bx+c= 0,aâ0aâ0are given by the formula:x=âb±b2â4ac2ax=âb±b2â4ac2a | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
How to solve a quadratic equation using the Quadratic Formula.Step 1.Write the quadratic equation in standard form,ax2+bx+c= 0. Identify the values ofa,b,c.Step 2.Write the Quadratic Formula. Then substitute in the values ofa,b,c.Step 3.Simplify.Step 4.Check the solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Write the quadratic equation in standard form,ax2+bx+c= 0. Identify the values ofa,b,c. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Write the Quadratic Formula. Then substitute in the values ofa,b,c. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Simplify. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 4.Check the solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Using the Discriminant,b2â 4ac, to Determine the Number and Type of Solutions of a Quadratic EquationFor a quadratic equation of the formax2+bx+c= 0,aâ0,aâ0,Ifb2â 4ac> 0, the equation has 2 real solutions.ifb2â 4ac= 0, the equation has 1 real solution.ifb2â 4ac< 0, the equation has 2 complex solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
For a quadratic equation of the formax2+bx+c= 0,aâ0,aâ0,Ifb2â 4ac> 0, the equation has 2 real solutions.ifb2â 4ac= 0, the equation has 1 real solution.ifb2â 4ac< 0, the equation has 2 complex solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Ifb2â 4ac> 0, the equation has 2 real solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
ifb2â 4ac= 0, the equation has 1 real solution. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
ifb2â 4ac< 0, the equation has 2 complex solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Methods to Solve Quadratic Equations:FactoringSquare Root PropertyCompleting the SquareQuadratic Formula | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Factoring | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Square Root Property | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Completing the Square | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Quadratic Formula | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
How to identify the most appropriate method to solve a quadratic equation.Step 1.Try Factoring first. If the quadratic factors easily, this method is very quick.Step 2.Try theSquare Root Propertynext. If the equation fits the formax2=kora(xâh)2=k, it can easily be solved by using the Square Root Property.Step 3.Use t... | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Try Factoring first. If the quadratic factors easily, this method is very quick. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Try theSquare Root Propertynext. If the equation fits the formax2=kora(xâh)2=k, it can easily be solved by using the Square Root Property. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Use theQuadratic Formula.Any other quadratic equation is best solved by using the Quadratic Formula. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
How to solve equations in quadratic form.Step 1.Identify a substitution that will put the equation in quadratic form.Step 2.Rewrite the equation with the substitution to put it in quadratic form.Step 3.Solve the quadratic equation foru.Step 4.Substitute the original variable back into the results, using the substitutio... | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Identify a substitution that will put the equation in quadratic form. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Rewrite the equation with the substitution to put it in quadratic form. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Solve the quadratic equation foru. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 4.Substitute the original variable back into the results, using the substitution. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 5.Solve for the original variable. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 6.Check the solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Methods to Solve Quadratic EquationsFactoringSquare Root PropertyCompleting the SquareQuadratic Formula | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Factoring | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Square Root Property | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Completing the Square | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Quadratic Formula | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
How to use a Problem-Solving Strategy.Step 1.Readthe problem. Make sure all the words and ideas are understood.Step 2.Identifywhat we are looking for.Step 3.Namewhat we are looking for. Choose a variable to represent that quantity.Step 4.Translateinto an equation. It may be helpful to restate the problem in one sentenc... | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Namewhat we are looking for. Choose a variable to represent that quantity. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 4.Translateinto an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 5.Solvethe equation using good algebra techniques. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Area of a TriangleFor a triangle with base,b, and height,h, the area,A, is given by the formulaA=12bh.A=12bh. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
For a triangle with base,b, and height,h, the area,A, is given by the formulaA=12bh.A=12bh. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Area of a RectangleFor a rectangle with length,L, and width,W, the area,A, is given by the formulaA=LW. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
For a rectangle with length,L, and width,W, the area,A, is given by the formulaA=LW. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Pythagorean TheoremIn any right triangle, whereaandbare the lengths of the legs, andcis the length of the hypotenuse,a2+b2=c2. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
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