Solving the Equation (x6)(x6) = 25
This equation presents a quadratic equation in a slightly disguised form. Let's break down the steps to solve it:
1. Expand the Equation
First, expand the left side of the equation by applying the FOIL method (First, Outer, Inner, Last):
(x6)(x6) = x²  6x  6x + 36 = x²  12x + 36
Now, our equation becomes: x²  12x + 36 = 25
2. Rearrange into Standard Quadratic Form
To solve a quadratic equation, it's crucial to have it in standard form: ax² + bx + c = 0.
Subtract 25 from both sides of the equation:
x²  12x + 36  25 = 0
Simplifying, we get: x²  12x + 11 = 0
3. Solve the Quadratic Equation
Now, we have a standard quadratic equation. There are several methods to solve this:

Factoring: Try to find two numbers that add up to 12 (the coefficient of the x term) and multiply to 11 (the constant term). In this case, the numbers are 1 and 11. Therefore, the equation can be factored as: (x1)(x11) = 0. This gives us two possible solutions: x = 1 and x = 11.

Quadratic Formula: If factoring doesn't seem straightforward, the quadratic formula provides a guaranteed solution for any quadratic equation. The formula is: x = (b ± √(b²  4ac)) / 2a
In our equation, a = 1, b = 12, and c = 11. Plugging these values into the formula, we get the same solutions as factoring: x = 1 and x = 11.
4. Verify the Solutions
To ensure the solutions are correct, substitute them back into the original equation:
 For x = 1: (16)(16) = (5)(5) = 25. This solution holds true.
 For x = 11: (116)(116) = (5)(5) = 25. This solution also holds true.
Therefore, the solutions to the equation (x6)(x6) = 25 are x = 1 and x = 11.