The document discusses various methods for solving quadratic equations: factoring, using square roots, and completing the square. It provides examples of using each method to solve equations as well as word problems involving quadratic equations. Factoring involves writing the quadratic as a product of two binomials and setting each factor equal to 0. Using square roots involves taking the square root of both sides of the equation to isolate the variable. Completing the square rewrites the equation by adding or subtracting terms to put it in the form (x+a)2=b and then taking the square root of both sides.

1. Solving Quadratic Equations Methods: Factoring Square Roots Completing the Square

2. Solutions of a Quadratic Equation Solve – find ALL values for the variable that make the equation true Which value(s) for the variable are solutions for the quadratic equation ? X = 3? X = 5? X = 0? X = -1? 3 ² - 4(3) = 5? 5 ²-4(5)=5? 9 – 12 = 5? 25-20=5? -3  5 5 = 5 3 is NOT 5 IS a solution a solution

3. Review: Factor a Quadratic Expression 1. Factor using the Greatest Common Factor . **GCF( ) 2. Factor using the Difference of Squares , if possible. **x²-y²=(x+y)(x-y) 3. Factor each Trinomial into 2 binomials, if possible. **ax²+bx+c=( )( ) FOIL or Box Method 4. Factor each quadratic completely. They may factor more than one time, so try to factor your answer again.

4. Factoring Practice Factor each expression completely: 1. x ²+x-72 2. 25-36x² 3. 4x³-4x 4. 21x³+28x²y² 5. 2x ²-5x+2 1. (x+9)(x-8) 2. (5-6x)(5+6x) 3. 4x(x+1)(x-1) 4. 7x ²y(3x+4y) 5. (2x-1)(x-2)

5. Solve by Factoring 1. Write the quadratic equation in standard form: ax²+bx+c=0 (Move all values to one side so the other side = 0) 2. Factor the quadratic expression completely (when two factors multiply to be 0, then at least one of them must =0) 3. Set each factor = 0 and solve. Example: 2x ² = 4x 2x² - 4x = 0 2x (x - 2) = 0 2x = 0 OR x – 2 = 0 X = 0 OR x = 2 Solutions/roots: {0, 2}

6. Solve by Factoring Examples 1. x ² - 9x = 0 2. 21 = 10x - x² 3. x² = 49 4. 4x² + 9 = 12x

7. Solve by Factoring Application Example #1 **The area of rectangle is 44 m². The length = 2x + 3 and the width = x . Find the length and width of the rectangle. 1. Write the formula for area of rectangle 2. Substitute into the formula Solve the quadratic equation. Check answer(s).

8. Solve by Factoring Application Example #2 **A ball is kicked into the air from the ground level. The height of the ball (h, in feet above the ground) at any time (t, in seconds after ball is kicked) can be modeled by the equation h(t) = -16t ² + 48t. How long is the ball in the air before it hits the ground? 1. What is the height of a ball on the ground? 2. Substitute the value of h into the equation. 3. Solve the quadratic equation and check answer(s).

9. Write a Quadratic Equation Given Roots If the roots of a quadratic equation are -2 and 6, write a quadratic model with leading coefficient of 1. Work backwards: start with your answer X = -2 or x = 6 solutions/roots x + 2 = 0 or x – 6 = 0 equation for roots (x + 2)(x - 6) = 0 quadratic equation x ² - 6x + 2x – 12 = 0 distribute using FOIL x² - 4x – 12 = 0 combine like terms This is the quadratic equation with roots {-2, 6}

10. Solve Quadratic by Square Roots Get the square term by itself: may be just x ² or an entire parentheses (x – 4) ². You will need to get rid of all numbers in front of and behind the squared term- use inverse operations. Square root both sides: you will have both a + AND – answer Solve each equation for x, the variable. They may be real or imaginary numbers. Example: 2( x – 5) ² = 24 (x – 5) ² = 12  X – 5 =   12 X – 5 =  2  3 X = 5  2  3 There are 2 solutions: X = 5 + 2  3, 5 - 2  3

11. Solve Quadratics by Square Roots Examples x ² = 64 2. (x + 7)² = 9 3. 3x² + 4 = -8 4. ¼(x – 1)²+12= -6

12. Solve Quadratic by Completing the Square 1. If x has a coefficient, divide by it on both sides (every term) x ²+bx+c=0 2. Write the x terms on one side and the constant on the other x ²+bx= -c 3. Take half of the coefficient of x, square it and add it to both sides x ²+ bx + (b/2)² = -c + (b/2)² 4. Write the factored form (x + b/2) ² = simplified # 5. Square root both sides to solve the equation. **Don’t forget + and – answers! Example: x ²-8x+3=0 x² - 8x = -3 x² - 8x + (-4) ² = -3 + (-4) ² (x - 4) ² = -3+16 (x - 4) ² = 13  X – 4 =  13 X = 4  13 This is 2 different solutions!

13. Solve Quadratic by Completing the Square Examples 1. x ²+6x=7 2. x²-4x+6=0 3. 3x²=6x-6 4. 2x²-2= -16x