2%2B(x%2B1)2%3D10.jpeg "(x-1)2+(x+1)2=10")
Solving the Equation (x-1)² + (x+1)² = 10
This equation is a quadratic equation in disguise. Let's break it down and find the solutions:
1. Expand the Squares
Begin by expanding the squares:
- (x-1)² = x² - 2x + 1
- (x+1)² = x² + 2x + 1
Substituting these back into the original equation, we get:
(x² - 2x + 1) + (x² + 2x + 1) = 10
2. Simplify the Equation
Combine like terms:
2x² + 2 = 10
3. Solve for x
- Subtract 2 from both sides: 2x² = 8
- Divide both sides by 2: x² = 4
- Take the square root of both sides: x = ±2
Therefore, the solutions to the equation (x-1)² + (x+1)² = 10 are x = 2 and x = -2.
Let's check our solutions:
- For x = 2: (2-1)² + (2+1)² = 1² + 3² = 1 + 9 = 10
- For x = -2: (-2-1)² + (-2+1)² = (-3)² + (-1)² = 9 + 1 = 10
Both solutions satisfy the original equation.
Key Points:
- This equation is a quadratic equation in disguise because it simplifies to a standard quadratic form after expanding the squares.
- The solutions represent the x-coordinates where the graph of the equation intersects the x-axis.
- Always check your solutions by plugging them back into the original equation to ensure accuracy.