%5E2-36%3D0.jpeg "(x+3)^2-36=0")
Solving the Equation (x+3)^2 - 36 = 0
This equation is a quadratic equation in disguise. Let's break down how to solve it:
1. Simplifying the Equation
- Expand the square: (x+3)^2 = (x+3)(x+3) = x^2 + 6x + 9
- Substitute: Now the equation becomes: x^2 + 6x + 9 - 36 = 0
- Combine like terms: x^2 + 6x - 27 = 0
2. Solving the Quadratic Equation
We have a standard quadratic equation now: ax^2 + bx + c = 0, where a = 1, b = 6, and c = -27. We can solve this using various methods:
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Factoring:
- Find two numbers that multiply to -27 and add up to 6. These numbers are 9 and -3.
- Factor the equation: (x + 9)(x - 3) = 0
- Set each factor to zero and solve:
- x + 9 = 0 => x = -9
- x - 3 = 0 => x = 3
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Quadratic Formula:
- The quadratic formula solves for x in any equation of the form ax^2 + bx + c = 0:
- x = (-b ± √(b^2 - 4ac)) / 2a
- Substitute the values:
- x = (-6 ± √(6^2 - 4 * 1 * -27)) / 2 * 1
- x = (-6 ± √(144)) / 2
- x = (-6 ± 12) / 2
- x = 3 or x = -9
- The quadratic formula solves for x in any equation of the form ax^2 + bx + c = 0:
3. The Solutions
Therefore, the solutions to the equation (x+3)^2 - 36 = 0 are x = 3 and x = -9.