# FACTORING QUADRATICS 8.1.1 And 8.1

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FACTORING QUADRATICS8.1.1 and 8.1.2Chapter 8 introduces students to quadratic equations. These equations can be written inthe form of y ax 2 bx c and, when graphed, produce a curve called a parabola.There are multiple methods that can be used to solve quadratic equations. One of themrequires students to factor.Students have used algebra tiles to build rectangles of quadratic expressions. Later, theyused the formula for the area of a rectangle, (base)(height) area , to create genericrectangles and write equations expressing the area as both a sum and a product. In thefigure below, the length and width of the rectangle are factors since they multiplytogether to produce the quadratic x 2 6x 8 . Notice the 4x and 2x are locateddiagonally from each other. They are like terms and can be combined and written as 6x .The dimensions of the rectangle are (x 2) and (x 4) . The area is x 2 6x 8 .44x8x22x xx 2The factors of x 2 6x 8 are (x 2) (x 4) .The ax2 term (1x 2 ) and the constant (8) are always diagonal to one another in a genericrectangle. In this example, the diagonal product is 8x2 and the c term is the product:2(4) 8 . The two x-terms are the other diagonal and can be combined into a sum whenthey are like terms. The b term of the quadratic is the sum of the factors: 2x 4x 6x .Their diagonal product is (2x)(4x) 8x 2 , the same as the other diagonal. (See the MathNotes box on page 332 for why the products of the two diagonals are always equal.) Tofactor, students need to think about sums and products simultaneously. Students canGuess & Check or use a “diamond problem” from Chapter 1 to help organize their sumsand products. See the Math Notes box on page 338.58 2006 CPM Educational Program. All rights reserved.Algebra Connections Parent Guide

Example 112Factor x 2 7x 12 .x2Sketch a generic rectangle with 4 sections.Write the x2 and the 12 along one diagonal.Find two terms whose product is 12x2 and whose sum is 7x(3x and 4x). Students are familiar with this situation as a“diamond problem” from Chapter 1.3 xWrite these terms as the other diagonal.3x12x24x3x12x24xx 4Find the base and height of the rectangle by using the partial areas.Write the complete equation.(x 3)(x 4) x 2 7x 12Example 2The same process works with negative values. Factor x 2 7x ! 30 .Sketch a generic rectangle with 4 sections.Write the x2 and the -30 along one diagonal.Find two terms whose product is -30x2 and whosesum is 7x.–3x!30x210xWrite these terms as the other diagonal.3–x–3x!30x210xx 10Find the base and height of the rectangle. Pay particular attention to the signs so the sum is 7x,not -7x.Write the complete equation.(x ! 3)(x 10) x 2 7x ! 30Chapter 8: Quadratics 2006 CPM Educational Program. All rights reserved.59

Example 3Factor x 2 ! 15x 56 .Sketch a generic rectangle with 4 sections.Write the x2 and the 56 along one diagonal.Find two terms whose product is 56x andwhose sum is -15x.2–8x56x2–7x8–x–8x56x2–7xx – 7Write these terms as the other diagonal.Find the base and height of the rectangle. Mentally multiply the factors to be sure they arecorrect. Pay close attention to the signs of the factors.Write the complete equation.(x ! 7)(x ! 8) x 2 ! 15x 56Example 4Factor 2x 2 7x 6 .Sketch a generic rectangle with 4 sections.Write the 2x2 and the 6 along one diagonal.Find two terms whose product is 12x2 andwhose sum is 7x.3x62x 24xWrite these terms as the other diagonal.3 3x 2x 2x 264xx 2Find the base and height of the rectangle.Write the complete equation.(2x 3)(x 2) 2x 2 7x 660 2006 CPM Educational Program. All rights reserved.Algebra Connections Parent Guide

Example 5Factor 12x 2 ! 19x 5 .Sketch a generic rectangle with 4 sections.Write the 12x2 and the 5 along one diagonal.Find two terms whose product is 60x2 andwhose sum is -19x.–15x512x 2– 4x5 –15x–4x 12x 25– 4x3x – 1Write these terms as the other diagonal.Find the base and height of the rectangle. Check the signs of the factors.Write the complete equation.(3x ! 1)(4x ! 5) 12x 2 ! 19x 5Problems1.x 2 5x 62.2x 2 5x 33.3x 2 4x 14.x 2 10x 255.x 2 15x 446.x 2 7x 67.x 2 11x 248.x 2 4x ! 329.4x 2 12x 910.12x 2 11x ! 511.x 2 x ! 7212.3x 2 ! 20x ! 713.x 2 ! 11x 28142x 2 11x ! 615.2x 2 5x ! 316.x 2 ! 3x ! 1017.4x 2 ! 12x 918.3x 2 2x ! 519.6x 2 ! x ! 220.9x 2 ! 18x 8Answers1.(x 2)(x 3)2.(x 1)(2x 3)3.(3x 1)(x 1)4.(x 5)(x 5)5.(x 11)(x 4)6.(x 6)(x 1)7.(x 8)(x 3)8.(x 8)(x ! 4)9.(2x 3)(2x 3)10.(3x ! 1)(4x 5)11.(x ! 8)(x 9)12.(x ! 7)(3x 1)13.(x ! 4)(x ! 7)14.(x 6)(2x ! 1)15.(x 3)(2x ! 1)16.(x ! 5)(x 2)17.(2x ! 3)(2x ! 3)18.(3x 5)(x ! 1)19.(2x 1)(3x ! 2) 20.Chapter 8: Quadratics 2006 CPM Educational Program. All rights reserved.(3x ! 4)(3x ! 2)61

USING THE ZERO PRODUCT PROPERTY8.2.3 and 8.2.4A parabola is a symmetrical curve. Its highest or lowest point is called the vertex. It isformed using the equation y ax 2 bx c . Students have been graphing parabolas bysubstituting values for x and solving for y. This can be a tedious process if the range ofx values is unknown. One possible shortcut is to find the x-intercepts first, then find thevertex and other convenient points. To find the x-intercepts, substitute 0 for y and solvethe quadratic equation, 0 ax 2 bx c . Students will learn multiple methods to solvequadratic equations in this course.The method described below utilizes two ideas:(1) When the product of two or more numbers is zero, then one of thenumbers must be zero.(2) Some quadratics can be factored into the product of two binomials,where coefficients and constants are integers.This procedure is called the Zero Product Method or Solving by Factoring. See theMath Notes boxes on pages 349 and 361.Example 1Find the x-intercepts of y x 2 6x 8 .The x-intercepts are located on the graphwhere y 0 , so write the quadraticexpression equal to zero, then solve for x.x 2 6x 8 0Factor the quadratic.(x 4)(x 2) 0Set each factor equal to 0.(x 4) 0 or (x 2) 0Solve each equation for x.x !4 or x !2The x-intercepts are (-4, 0) and (-2, 0).You can check your answers by substituting them into the original equation.(!4)2 6(!4) 8 " 16 ! 24 8 " 0(!2)2 6(!2) 8 " 4 ! 12 8 " 062 2006 CPM Educational Program. All rights reserved.Algebra Connections Parent Guide

Example 2Solve 2x 2 7x ! 15 0.Factor the quadratic.(2x ! 3)(x 5) 0Set each factor equal to 0.(2x ! 3) 0 or (x 5) 0Solve for each x.2x 3x 23 or x !5Example 3If the quadratic does not equal 0, rewrite it algebraically so that it does, then use the zero productproperty.Solve 2 6x 2 ! x .Set the equation equal to 0.2 6x 2 ! x0 6x 2 ! x ! 2Factor the quadratic.Solve each equation for x.0 (2x 1)(3x ! 2)(2x 1) 0!!!!!!!or!!!!!!!(3x ! 2) 02x !1!!!!!or!!!!!!!!!!!!!!!3x 2x ! 12 !!!!!!!!!!!!!!!!!!!!!!!!!x 23Example 4Solve 9x 2 ! 6x 1 0 .Factor the quadratic.Solve each equation for x. Notice thefactors are the same so there will beonly one x value. For this parabola, thex-intercept is the vertex.Chapter 8: Quadratics 2006 CPM Educational Program. All rights reserved.9x 2 ! 6x 1 0(3x ! 1)(3x ! 1) 0(3x ! 1) 03x 1x 1363

ProblemsSolve for x.1.x 2 ! x ! 12 02.3x 2 ! 7x ! 6 03.x 2 x ! 20 04.3x 2 11x 10 05.x 2 5x !46.6x ! 9 x 27.6x 2 5x ! 4 08.x 2 ! 6x 8 09.6x 2 ! x ! 15 010.4x 2 12x 9 011.x 2 ! 12x 2812.2x 2 8x 6 013.2 9x 5x 214.2x 2 ! 5x 315.x 2 45 ! 4xAnswers1.4 or !32.! 23 or 33.4.! 53 or !25.!4 or !16.7.! 43 or8.4 or 29.12!5 or 43! 23 or5310.! 2311.14 or !212.!1 or !313.! 15 or 214.! 12 or 315.5 or -964 2006 CPM Educational Program. All rights reserved.Algebra Connections Parent Guide

USING THE QUADRATIC FORMULA8.3.1When a quadratic equation is not factorable, another method is needed to solve for x. TheQuadratic Formula can be used to calculate the roots of the equation, that is, thex-intercepts of the quadratic. The Quadratic Formula can be used with any quadraticequation, factorable or not. There may be two, one or no solutions, depending onwhether the parabola intersects the x-axis twice, once, or not at all.!b b 2 ! 4ac. The symbol is read as “plus2aor minus.” It is shorthand notation that tells you to calculate the formula twice, once with and again with – to get both x-values.For any quadratic ax 2 bx c 0 , x To use the formula, the quadratic equation must be written in standard form:ax 2 bx c 0 . This is necessary to correctly identify the values of a, b, and c. Oncethe equation is in standard form and equal to 0, a is the coefficient of the x 2 term, b isthe coefficient of the x term and c is the constant. See the Math Notes boxes on pages 358and 361 and problem 12-52 on page 510.Example 1Solve 2x 2 ! 5x ! 3 0 .Identify a, b, and c. Watch your signs carefully.a 2,!b !5,!c !3Write the quadratic formula.x !b b 2 ! 4ac2aSubstitute a, b, and c in the formula and do theinitial calculations.x !(!5) (!5)2 ! 4(2)(!3)2(2)x 5 25 ! (!24)4x 5 494x 5 74Simplify the.Calculate both values of x. 12 3 or x 45! 74 !24 ! 12The solution is x 3 or x ! 12 . The x-intercepts are (3, 0) and ( ! 12 , 0).Chapter 8: Quadratics 2006 CPM Educational Program. All rights reserved.65

Example 2Solve 3x 2 5x 1 0 .Identify a, b, and c.a 3,!b 5,!c 1Write the quadratic formula.x !b b 2 ! 4ac2aSubstitute a, b, and c in the formula and do theinitial calculations.x !(5) (5)2 ! 4(3)(1)2(3)x !5 25 ! 126x !5 136Simplify the.!5 13!5 ! 13or x . These are the exact values of x.66The x values are the x-intercepts of the parabola. To graph these points, use a calculator to findthe decimal approximation of each one. The x-intercepts are ( ! "0.23 , 0) and ( ! "1.43 , 0).The solution is x Example 3Solve 25x 2 ! 20x 4 0 .Identify a, b, and c.a 25,!b !20,!c 4Write the quadratic formula.x !b b 2 ! 4ac2aSubstitute a, b, and c in the formula and do theinitial calculations.x !(!20) (!20)2 ! 4(25)(4)2(25)x 20 400 ! 40050x 20 050Simplify the.This quadratic has only one solution: x 25 .66 2006 CPM Educational Program. All rights reserved.Algebra Connections Parent Guide

Example 4Solve x 2 4x 10 0 .Identify a, b, and c.a 1,!b 4,!c 10Write the quadratic formula.x !b b 2 ! 4ac2aSubstitute a, b, and c in the formula and do theinitial calculations.x !(4) (4)2 ! 4(1)(10)2(1)x !4 16 ! 402x !4 !242Simplify the.It is impossible to take the square root of a negative number; therefore this quadratic has no realsolution. The graph would still be a parabola, but there would be no x-intercepts. This parabolais above the x-axis.Example 5Solve (3x 1)(x 2) 1 .Rewrite in standard form.(3x 1)(x 2) 13x 2 7x 2 13x 2 7x 1 0Identify a, b, and c.a 3,!b 7,!c 1Write the quadratic formula.x !b b 2 ! 4ac2aSubstitute a, b, and c in the formula and do theinitial calculations.x !(7) (7)2 ! 4(3)(1)2(3)x !7 49 ! 126x !7 376Simplify .The x-intercepts are (! "0.15,!0) and (! "2.18,!0) .Chapter 8: Quadratics 2006 CPM Educational Program. All rights reserved.67

Example 6Solve 3x 2 6x 1 0 .Identify a, b, and c.a 3,!b 6,!c 1Write the quadratic formula.x !b b 2 ! 4ac2aSubstitute a, b, and c in the formula and do theinitial calculations.x !(6) (6)2 ! 4(3)(1)2(3)x !6 36 ! 126x !6 246Simplify .The x-intercepts are ( ! "1.82 , 0) and ( ! "0.18 , 0).Math Note 8.3.3 describes another form of this expression that can be written by simplifying thesquare root. The result is equivalent to the exact values above.Factor the24 , then simplify by taking thesquare root of 4.24 4 6 2 6x !6 2 66x !3 63Simplify the fraction by dividing every term by 2.68 2006 CPM Educational Program. All rights reserved.Algebra Connections Parent Guide

ProblemsSolve each of these using the Quadratic Formula.1.x2 ! x ! 2 02.x2 ! x ! 3 03.!3x 2 2x 1 04.!2 ! 2x 2 4x5.7x 10 ! 2x 26.!6x 2 ! x 6 07.6 ! 4x 3x 2 88.4x 2 x ! 1 09.x 2 ! 5x 3 010.0 10x 2 ! 2x 311.x(!3x 5) 7x ! 1012.(5x 5)(x ! 5) 7xAnswers1.x 2or x !12.x 1 2 133.x ! 13 or x 16.145x 1 !12! 2.30!or!"1.304.x !15.x !7 1294! 1.09!or!"4.597.x 4 406 2 1038.! 1.72!or!"0.3910.No solutionx !1 178! "1.09!or!0.929.! 0.39!or!"0.6411.x 2 124!6 1 !331! "2.19!or!1.52Chapter 8: Quadratics 2006 CPM Educational Program. All rights reserved.x 5 132! 4.30!or!0.7012.x 27 122910! 6.21!or!"0.8169

The method described below utilizes two ideas: (1) When the product of two or more numbers is zero, then one of the numbers must be zero. (2) Some quadratics can be factored into the product of two binomials, where coefficients and constants are integers. This procedure is called the Zero Product Method or Solving by Factoring. See the

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