(x+3)2 = 64 Responses

Jasonbradley

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Jun 16, 2024

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Solving the Equation (x + 3)² = 64

This equation is a quadratic equation in disguise, and we can solve it using a few different methods:

Method 1: Square Root Property

1. Take the square root of both sides: √(x + 3)² = ±√64

2. Simplify: x + 3 = ±8

3. Solve for x:

   • x + 3 = 8 => x = 5
   • x + 3 = -8 => x = -11

Therefore, the solutions to the equation (x + 3)² = 64 are x = 5 and x = -11.

Method 2: Expanding and Solving

1. Expand the left side: x² + 6x + 9 = 64

2. Move all terms to one side: x² + 6x - 55 = 0

3. Factor the quadratic: (x + 11)(x - 5) = 0

4. Set each factor to zero and solve:

   • x + 11 = 0 => x = -11
   • x - 5 = 0 => x = 5

Again, we arrive at the solutions x = 5 and x = -11.

Verifying the Solutions

We can plug our solutions back into the original equation to verify they are correct:

• For x = 5: (5 + 3)² = 8² = 64 (True)
• For x = -11: (-11 + 3)² = (-8)² = 64 (True)

Both solutions hold true for the original equation.

In summary, the solutions to the equation (x + 3)² = 64 are x = 5 and x = -11.