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Finding the Solutions, x- intercepts, Roots, or Zeros of A Quadratic

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x -intercept(s): Where the graph of y = ax 2 + bx + c crosses the x -axis. The value(s) for x that makes a quadratic equal 0. Solution(s) OR Roots: The value(s) of x that satisfies 0 = ax 2 + bx + c. Zeros: The value(s) of x that make ax 2 + bx + c equal 0. x –Intercepts, Solutions, Roots, and Zeros in Quadratics

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Zero Product Property If a. b = 0, then a and or b is equal to 0 Ex: Solve the following equation below. 0 = ( x + 14 )( 6x + 1 ) Would you rather solve the equation above or this: 0 = 6x x + 14 ?

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30x 14x (2x 2 )(-12) 420x x 2 ax 2 c ax2 ax2 Example ___ bx 44x 30x 14x c Product Sum 2x2x 6x6x7 5 GCF Solve: Factor to rewrite as a product Use the Zero-Product Property Solve for 0 first!

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(x 2 )(-7) -7x 2 -7 x2x2 ax 2 c ax2 ax2 Example ___ bx -3x IMPOSSIBLE c Product Sum Use the Zero Product Property to find the roots of: But this parabola has two zeros. Just because a quadratic is not factorable, does not mean it does not have roots. Thus, there is a need for a new algebraic method to find these roots.

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Quadratic Formula For ANY 0 = ax 2 + bx +c (standard form) the value(s) of x is given by: Opposite of b “All Over” MUST equal 0 Plus or Minus This formula will provide the solutions (or lack thereof) to ANY Quadratic.

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Example Solve: Solve for 0 first! a = b = c = Find the values of “ a,” “ b,” “ c ” Substitute into the Quadratic Formula Simplify the expression in the square root first The square root can be simplified. Since the answers will be rational, it is best to list both. Or

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