(x-5)%3D8.jpeg "(x-3)(x-5)=8")
Solving the Equation (x-3)(x-5) = 8
This equation represents a quadratic equation in disguise. Let's break it down step by step to find the solutions for x.
Expanding the Equation
First, we need to expand the left side of the equation by multiplying the terms:
(x-3)(x-5) = 8
x² - 5x - 3x + 15 = 8
x² - 8x + 15 = 8
Rearranging to Standard Form
Next, we rearrange the equation to bring it to the standard quadratic form (ax² + bx + c = 0):
x² - 8x + 15 - 8 = 0
x² - 8x + 7 = 0
Solving the Quadratic Equation
Now we can solve the equation using various methods, such as factoring, the quadratic formula, or completing the square. Let's use factoring in this case:
We need to find two numbers that add up to -8 and multiply to 7. Those numbers are -1 and -7:
(x - 1)(x - 7) = 0
For the product of two terms to be zero, at least one of them must be zero. Therefore, we have two possible solutions:
x - 1 = 0 or x - 7 = 0
x = 1 or x = 7
Conclusion
Therefore, the solutions to the equation (x-3)(x-5) = 8 are x = 1 and x = 7.