(x-6)%3D25.jpeg "(x-6)(x-6)=25")
Solving the Equation (x-6)(x-6) = 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):
(x-6)(x-6) = 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:
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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: (x-1)(x-11) = 0. This gives us two possible solutions: x = 1 and x = 11.
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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: (1-6)(1-6) = (-5)(-5) = 25. This solution holds true.
- For x = 11: (11-6)(11-6) = (5)(5) = 25. This solution also holds true.
Therefore, the solutions to the equation (x-6)(x-6) = 25 are x = 1 and x = 11.