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Solving the Equation (x-5)(x+1) = 7
This article will guide you through the steps to solve the quadratic equation (x-5)(x+1) = 7.
1. Expand the Equation
First, we need to expand the left side of the equation by multiplying the factors:
(x-5)(x+1) = 7
- x² - 4x - 5 = 7
2. Rearrange to Standard Quadratic Form
Next, we need to move the constant term from the right side to the left side to get the equation in standard quadratic form (ax² + bx + c = 0):
- x² - 4x - 5 - 7 = 0
- x² - 4x - 12 = 0
3. Solve the Quadratic Equation
Now we have a standard quadratic equation and can solve for x using one of the following methods:
- Factoring: Try to find two numbers that multiply to -12 and add up to -4. The numbers -6 and 2 satisfy these conditions. Therefore, we can factor the equation as:
- (x - 6)(x + 2) = 0
- This gives us two possible solutions:
- x - 6 = 0 => x = 6
- x + 2 = 0 => x = -2
- Quadratic Formula: If factoring is not easily possible, use the quadratic formula:
- x = (-b ± √(b² - 4ac)) / 2a
- Where a = 1, b = -4, and c = -12
- Substitute these values into the formula and solve for x. This will also give you the solutions x = 6 and x = -2.
Conclusion
Therefore, the solutions to the equation (x-5)(x+1) = 7 are x = 6 and x = -2.