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Quadratics

Factoring Quadratic Equations

Problems

Problems

A quadratic equation is an equation of the form ax 2 + bx + c = 0 , where a≠ 0 , and a , b , and c are real numbers.

Solving Quadratic Equations by Factoring

We can often factor a quadratic equation into the product of two binomials. We are then left with an equation of the form (x + d )(x + e) = 0 , where d and e are integers.

The zero product property states that, if the product of two quantities is equal to 0 , then at least one of the quantities must be equal to zero. Thus, if (x + d )(x + e) = 0 , either (x + d )= 0 or (x + e) = 0 . Consequently, the two solutions to the equation are x = - d and x = - e .

Example 1: Solve for x : x 2 - 5x - 14 = 0

x 2 - 5x - 14 = (x - 7)(x + 2) = 0
x - 7 = 0 or x + 2 = 0
x = 7 or x = - 2

Thus, the solution set is { -2, 7} .

Example 2: Solve for x : x 2 + 6x + 5 = 0

x 2 + 6x + 5 = (x + 1)(x + 5) = 0
x + 1 = 0 or x + 5 = 0
x = - 1 or x = - 5

Thus, the solution set is { -1, -5} .

Example 3: Solve for x : 2x 2 - 16x + 24 = 0

2x 2 -16x + 24 = 2(x 2 - 8x + 12) = 2(x - 2)(x - 6) = 0
x - 2 = 0 or x - 6 = 0
x = 2 or x = 6

Thus, the solution set is {2, 6} .

Example 4: Solve for x : x 2 + 6x + 9 = 0

x 2 +6x + 9 = (x + 3)(x + 3) = (x + 3)2 = 0
x + 3 = 0
x = - 3

Thus, the solution set is { -3} .

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