7.2 Solving Quadratic Equations by Factoring

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1 7.2 Solving Quadratic Equations by Factoring 1

2 Factoring Review There are four main types of factoring: 1) Removing the Greatest Common Factor 2) Difference of square a 2 b 2 3) Trinomials in the form x 2 + bx + c 4) Trinomials in the form ax 2 + bx + c 1. Removing the GCF Determine the GCF of each term. This is the GCF of the numbers and the lowest exponent of the variables. This GCF goes outside the brackets. Divide each term by the GCF. This quotient goes inside the brackets. Example: Factor each of the following; A) 5x 15 B) 3x 3 + 6x 2 21x 2

3 2. Difference of Squares a 2 b 2 Write ( + )( ) Take the square root of each term. The square root of the first term is the first term in the brackets. The square root of the second term is the last terms in the brackets. Example: Factor each of the following: A) x 2 81 B) 9x

4 3. Trinomials of form x 2 + bx + c Write ( x )( x ) Determine which two numbers multiplied together gives you c and added together gives you b. These factors go in the empty positions. Example: Factor each of the following completely: A) x 2 + 9x + 20 B) x 2 12x + 20 C) x 2 5x 24 D) x 2 + 3x 10 4

5 4. Trinomials of form ax 2 + bx + c Multiply the numerical coefficients in the first and last terms. Determine which two multiplied together gives you this product and added together gives you b. Rewrite the trinomial by breaking up the middle term into these two factors Then use the box method to factor Example: Factor each of the following: A) 6x x + 5 B) 3x 2 + 4x 4 5

6 Solving Quadratic Equations by Factoring When the product of two or more factors is zero then one of the factors must be zero. We will use this fact to solve quadratic equations. Steps to Solve Quadratic Equations by Factoring 1. Move all the terms to one side so that one side equals 0. Remember if you change sides you change signs. 2. Factor the non zero side completely. 3. Let all binomial and monomial factors equal Solve for the variable in each case. 6

7 Example: Solve each of the following equations: A) 12x x = 0 B) 12x 2 = 8x E) C) 8x 2 18 = 0 D) F) x 2 + 3x = 18 7

8 G) E) 2x 2 20x + 48 = 0 F) H) 6x 2 11x = 10 8

9 Ex 2. Find the roots of the equation 2x 2 4 = 0 using the square root property. Ex 3. Find the roots of the equation x 2 8 = 0 using the square root property. 9

10 Using Zeroes to Express a Quadratic Equation Goal: Develop a quadratic equation/function given a root/zero In the previous unit, you had to determine a unique parabola given the x intercepts and another point. Now you will write a quadratic equation based only on the x intercepts. Since you will only be given the x intercepts, multiple quadratic equations will exist. Ex 4. Use your calculator to graph the following functions. What do the graphs have in common 10

11 Ex 5. Write two different quadratic equations in standard form having roots 3 and 3. Ex 6. Determine the quadratic function y = ax 2 + bx + c that has zeroes 2 and 6. 11

12 Example: An osprey dives toward the water to catch a salmon. Its height above the water, in metres, t seconds after it begins its dive, is approximated by. Algebraically determine the time it takes for the osprey to reach a return height of 20 m. 12

13 Example: A travel agency has 16 people signed up for a trip. The revenue for the trip is modeled by the function R(x) = 100x x where x represents the number of additional people to sign up. How many additional people must sign up for the revenue to reach $40000? P , 13

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A quadratic expression is a mathematical expression that can be written in the form 2

A quadratic expression is a mathematical expression that can be written in the form 2 118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is