10.4 Solving Equations in Factored Form

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Presentation transcript:

10.4 Solving Equations in Factored Form Algebra 10.4 Solving Equations in Factored Form

Quadratic Equations in Standard Form You have already learned the standard form of a quadratic equation: ax² + bx + c = 0 where a ≠ 0

Quadratic Equations in Factored Form Consider the equation: x² + 8x + 15 = 0 Here is the same equation, in factored form: (x + 5)(x + 3) = 0 FOIL to be sure x² +3x +5x +15 x² + 8x + 15 = 0

Two Ways to Solve: x² + 8x + 15 = 0 Quadratic Formula Factored form (x + 5)(x + 3) = 0 -8 ± 8² - 4(1)(15) √ x = 2(1) This means that for the equation to be true, either: -8 ± √ 4 x = or x + 5 = 0 x + 3 = 0 2 -5 -5 -3 -3 -8 ± 2 -6 -10 x = = x = -5 x = -3 2 2 2 x = -3, -5 x = -3, -5

ZERO PRODUCT PROPERTY If a product equals zero, then one of the factors must be 0. If ab = 0, then either a = 0 b = 0 or both = 0

Apply the ZERO PRODUCT PROPERTY…… We said that the solutions to the equation x² + 8x + 15 = 0 or (x +5)(x + 3) are -3 and -5. Substitute each solution into the equation in factored form: (-3 + 5)(-3 + 3) = 0 Substitute -3 for x (2) (0) = 0 TRUE (-5 + 5)(-5 + 3) = 0 Substitute -5 for x (0) (-2) = 0 TRUE Substitute into original equation: (-3)² + 8(-3) + 15 = 0 True (-5)² + 8(-5) + 15 = 0 True

Try these (x + 7) (x – 3) = 0 x = -7 or x = 3 (x - 4)(x – 5) = 0 +5 +5 +3 +3 2x = 5 x = 3 2 2 5 x = x = 3 2

Shortcut: (x + 7) (x – 3) = 0 -7 +3 When the factors are in the form (x +__ ) or (x - __ ) the solutions will always be the opposite of the factors, in this case -7 and +3. (x + 7) (x – 3) = 0 Ask yourself: what value of x will make this binomial = 0? Ask yourself: what value of x will make this binomial = 0? -7 +3

Shortcut: (3 + 2) (3x + 2) (x – 1) = 0 The other solution is +1 -2 = 0 Ask yourself: what value of x will make this binomial = 0? The other solution is +1 -2 (3 + 2) = 0 3 The denominator would cancel with 3 when multiplied The numerator would cancel with +2 when added 2 One solution is - 3

Try these x = -½ or x = 3 (2x + 1) (x – 3) = 0 (3x - 4)(2x + 5) = 0

A Few together from the HW P. 600 #29, 43

Homework pg. 600 # 19 - 46, 69-77