(x+7)^2-9=0

2 min read Jun 17, 2024

Solving the Equation (x+7)^2 - 9 = 0

This equation is a quadratic equation in disguise, meaning it can be rewritten in the standard form of ax² + bx + c = 0. Let's break down the steps to solve it:

1. Expand the Square

First, expand the squared term: (x+7)² = (x+7)(x+7) = x² + 14x + 49

Now the equation becomes: x² + 14x + 49 - 9 = 0

2. Simplify

Combine the constant terms: x² + 14x + 40 = 0

3. Solve the Quadratic Equation

Now we have a standard quadratic equation. There are several ways to solve this:

Factoring:
- Find two numbers that add up to 14 and multiply to 40 (in this case, 10 and 4).
- Rewrite the equation: (x + 10)(x + 4) = 0
- Set each factor equal to zero and solve:
  - x + 10 = 0 => x = -10
  - x + 4 = 0 => x = -4
Quadratic Formula:
- The quadratic formula solves for x in any equation of the form ax² + bx + c = 0:
  - x = [-b ± √(b² - 4ac)] / 2a
- In this case, a = 1, b = 14, and c = 40.
- Substitute the values into the formula and solve for x:
  - x = [-14 ± √(14² - 4 * 1 * 40)] / (2 * 1)
  - x = [-14 ± √(196 - 160)] / 2
  - x = [-14 ± √36] / 2
  - x = (-14 ± 6) / 2
  - x = -10 or x = -4

Conclusion

Therefore, the solutions to the equation (x+7)² - 9 = 0 are x = -10 and x = -4.