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## Factor and Solve Quadratic Equations

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**Factor and SolveQuadratic Equations**Ms. Nong**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**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**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**Vertex (h,k)**• Maximum point if the parabola is up-side-down • Minimum point is when the Parabola is UP a>0 a<0**Can you answer these questions?**• How many Roots? • Where is the Vertex? (Maximum or minimum) • What is the Y-Intercepts?**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**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**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**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)**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**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**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