# Factor and Solve Quadratic Equations

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## Factor and Solve Quadratic Equations

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1. What is in this unit? • Graphing the Quadratic Equation • Identify the vertex and intercept(s) for a parabola • Solve by taking SquareRoot & Squaring • Solve by using the Quadratic Formula • Solve by Completing the Square • Factor & Solve Trinomials (split the middle) • Factor & Solve DOTS: difference of two square • Factor GCF (greatest common factors) • Factor by Grouping

2. Parts of a Parabola • The ROOTS (or solutions) of a polynomial are its x-intercepts • Recall: The x-intercepts occur where y = 0. Roots ~ X-Intercepts ~ Zeros means the same

3. Solving a Quadratic • The x-intercepts (when y = 0)of a quadratic function are the solutions to the related quadratic equation. The number of real solutions is at most two. • One solution • X = 3 • Two solutions • X= -2 or X = 2 • No solutions

4. Vertex (h,k) • Maximum point if the parabola is up-side-down • Minimum point is when the Parabola is UP a>0 a<0

5. All parts labeled

6. Can you answer these questions? • How many Roots? • Where is the Vertex? (Maximum or minimum) • What is the Y-Intercepts?

7. What is in this unit? • Graph the quadratic equations (QE) • Solve by taking SquareRoot & Squaring • Solve by using the Quadratic Formula • Solve by Completing the Square • Factor & Solve Trinomials (split the middle) • Factor & Solve DOTS: difference of two square • Factor GCF (greatest common factors) • Factor by Grouping

8. Finding the Axis of Symmetry When a quadratic function is in standard form y = ax2 + bx + c, the equation of the Axis of symmetry is This is best read as … ‘the opposite of b divided by the quantity of 2 times a.’ Find the Axis of symmetry for y = 3x2 – 18x + 7 • The Axis of symmetry is x = 3. a = 3 b = -18

9. Finding the Vertex The Axis of symmetry always goes through the _______. Thus, the Axis of symmetry gives us the ____________ of the vertex. Vertex X-coordinate Find the vertex of y = -2x2 + 8x - 3 STEP 1: Find the Axis of symmetry The x-coordinate of the vertex is 2 a = -2 b = 8

10. Finding the Vertex Find the vertex of y = -2x2 + 8x - 3 STEP 1: Find the Axis of symmetry STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex. The vertex is (2 , 5)

11. y x y • x 2 3 Graphing a Quadratic Function STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. –1 5

12. Y-axis y x Y-intercept of a Quadratic Function The y-intercept of a Quadratic function can Be found when x = 0. The constant term is always the y- intercept

13. Example: Graph y= -.5(x+3)2+4 • a is negative (a = -.5), so parabola opens down. • Vertex is (h,k) or (-3,4) • Axis of symmetry is the vertical line x = -3 • Table of values x y -1 2 -2 3.5 -3 4 -4 3.5 -5 2 • Vertex (-3,4) • (-4,3.5) • (-2,3.5) • (-5,2) • (-1,2) • x=-3