%5E2-10%3D0.jpeg "(x-5)^2-10=0")
Solving the Quadratic Equation: (x-5)² - 10 = 0
This article will guide you through solving the quadratic equation (x-5)² - 10 = 0. We'll break down the process step-by-step, making it easy to understand.
1. Expanding the Equation
First, we need to expand the equation by squaring the term (x-5):
(x-5)² = (x-5)(x-5) = x² - 10x + 25
Now, the equation becomes:
x² - 10x + 25 - 10 = 0
Simplifying it further:
x² - 10x + 15 = 0
2. Using the Quadratic Formula
The quadratic formula is a standard method for solving equations in the form of ax² + bx + c = 0. In our equation, a = 1, b = -10, and c = 15.
The quadratic formula is:
x = [-b ± √(b² - 4ac)] / 2a
Let's substitute the values:
x = [10 ± √((-10)² - 4 * 1 * 15)] / (2 * 1)
Simplifying:
x = [10 ± √(100 - 60)] / 2
x = [10 ± √40] / 2
x = [10 ± 2√10] / 2
3. Finding the Solutions
Finally, we have two possible solutions for x:
x = (10 + 2√10) / 2 = 5 + √10
x = (10 - 2√10) / 2 = 5 - √10
Therefore, the solutions for the equation (x-5)² - 10 = 0 are x = 5 + √10 and x = 5 - √10.