(x%2B1)%3D(x-3)(x%2B3).jpeg "(2x-9)(x+1)=(x-3)(x+3)")
Solving the Equation: (2x-9)(x+1)=(x-3)(x+3)
This equation involves expanding brackets and simplifying to find the solution for x. Let's break down the steps:
1. Expand the Brackets
First, we expand the brackets on both sides of the equation using the FOIL method (First, Outer, Inner, Last):
- Left side: (2x-9)(x+1) = 2x² + 2x - 9x - 9
- Right side: (x-3)(x+3) = x² - 9 (This is a special case of the difference of squares)
Now the equation becomes: 2x² - 7x - 9 = x² - 9
2. Simplify and Solve for x
To solve for x, we need to rearrange the equation so that all terms are on one side:
- Subtract x² from both sides: x² - 7x - 9 = -9
- Add 9 to both sides: x² - 7x = 0
Now we have a quadratic equation. We can solve it by factoring:
- Factor out x: x(x - 7) = 0
For the product of two terms to be zero, at least one of the terms must be zero. Therefore, we have two possible solutions:
- x = 0
- x - 7 = 0 => x = 7
3. Verification
To ensure our solutions are correct, we can substitute them back into the original equation:
- For x = 0: (2(0)-9)(0+1) = (0-3)(0+3) => -9 = -9 (This holds true)
- For x = 7: (2(7)-9)(7+1) = (7-3)(7+3) => 56 = 56 (This holds true)
Therefore, the solutions to the equation (2x-9)(x+1)=(x-3)(x+3) are x = 0 and x = 7.