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

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**x–Intercepts, Solutions, Roots, and Zeros in Quadratics**

x-intercept(s): Where the graph of y = ax2 + 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 = ax2 + bx + c. Zeros: The value(s) of x that make ax2 + bx + c equal 0.

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**If a . b = 0, then a and or b is equal to 0**

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 = 6x2 + 85x + 14 ?

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**Use the Zero-Product Property**

Example Solve: Factor to rewrite as a product Product c (2x2)(-12) 420x2 5 Solve for 0 first! 30x 14x 35 ax2c 12x2 2x 30x x GCF ___ ax2 bx 6x 7 44x Use the Zero-Product Property Sum

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**But this parabola has two zeros.**

Example Use the Zero Product Property to find the roots of: Product But this parabola has two zeros. (x2)(-7) -7x2 c -7 ax2c IMPOSSIBLE x2 bx ___ ax2 -3x Sum 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 = ax2 + bx +c (standard form) the value(s) of x is given by: MUST equal 0 Plus or Minus Opposite of b “All Over” This formula will provide the solutions (or lack thereof) to ANY Quadratic.

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**Example a = b = c = 1 -3 -4 Solve: Find the values of “a,” “b,” “c”**

Solve for 0 first! a = b = c = 1 -3 -4 Simplify the expression in the square root first The square root can be simplified. Substitute into the Quadratic Formula Or Since the answers will be rational, it is best to list both.

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Factor: Factor: 1. s 2 r 2 – 4s 4 1. s 2 r 2 – 4s 4 2. 32b 4 - 48b 3 c + 18b 2 c 2 2. 32b 4 - 48b 3 c + 18b 2 c 2 3. xy + 3x – 2y - 6 3. xy + 3x – 2y -

Factor: Factor: 1. s 2 r 2 – 4s 4 1. s 2 r 2 – 4s 4 2. 32b 4 - 48b 3 c + 18b 2 c 2 2. 32b 4 - 48b 3 c + 18b 2 c 2 3. xy + 3x – 2y - 6 3. xy + 3x – 2y -

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