%5E2-9(x%2B1)%5E2%3D0.jpeg "(x-1)^2-9(x+1)^2=0")
Solving the Equation: (x-1)^2 - 9(x+1)^2 = 0
This equation is a quadratic equation in disguise. Let's break it down and find the solution(s):
1. Expand the Squares
First, we need to expand the squares using the formula (a - b)^2 = a^2 - 2ab + b^2 and (a + b)^2 = a^2 + 2ab + b^2.
- (x - 1)^2 = x^2 - 2x + 1
- 9(x + 1)^2 = 9(x^2 + 2x + 1) = 9x^2 + 18x + 9
Now our equation becomes: x^2 - 2x + 1 - 9x^2 - 18x - 9 = 0
2. Simplify the Equation
Combining like terms, we get: -8x^2 - 20x - 8 = 0
We can simplify this equation by dividing both sides by -4: 2x^2 + 5x + 2 = 0
3. Solve the Quadratic Equation
Now we have a standard quadratic equation in the form ax^2 + bx + c = 0. We can solve this using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 2, b = 5, and c = 2. Plugging these values into the quadratic formula, we get:
x = (-5 ± √(5^2 - 4 * 2 * 2)) / (2 * 2) x = (-5 ± √9) / 4 x = (-5 ± 3) / 4
This gives us two possible solutions:
- x = (-5 + 3) / 4 = -1/2
- x = (-5 - 3) / 4 = -2
Conclusion
Therefore, the solutions to the equation (x-1)^2 - 9(x+1)^2 = 0 are x = -1/2 and x = -2.