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(x+3)^2-16

2 min read Jun 16, 2024
(x+3)^2-16

Factoring and Solving (x+3)^2 - 16

The expression (x+3)^2 - 16 represents a quadratic equation in a slightly disguised form. Let's explore how to factor and solve it.

Factoring the Expression

  1. Recognize the Difference of Squares: The expression fits the pattern of a difference of squares: a^2 - b^2 = (a + b)(a - b)

  2. Identify 'a' and 'b': In our case, a = (x + 3) and b = 4.

  3. Apply the Formula: (x + 3)^2 - 16 = [(x + 3) + 4][(x + 3) - 4]

  4. Simplify: (x + 3)^2 - 16 = (x + 7)(x - 1)

Solving for x

To find the values of x that make the expression equal to zero, we set the factored expression equal to zero:

(x + 7)(x - 1) = 0

This means either (x + 7) = 0 or (x - 1) = 0

Solving for x in each case:

  • x + 7 = 0 => x = -7
  • x - 1 = 0 => x = 1

Therefore, the solutions to the equation (x + 3)^2 - 16 = 0 are x = -7 and x = 1.

Summary

We successfully factored the expression (x + 3)^2 - 16 into (x + 7)(x - 1) by recognizing the difference of squares pattern. This factorization allowed us to solve for the roots of the equation, finding the values of x that make the expression equal to zero.

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