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Solving the Quadratic Equation: (x-1)(x-3) = 15
This article will guide you through the steps of solving the quadratic equation (x-1)(x-3) = 15.
Expanding the Equation
First, we need to expand the left side of the equation by multiplying the terms:
(x-1)(x-3) = x² - 4x + 3
Now, our equation becomes:
x² - 4x + 3 = 15
Rearranging the Equation
To solve for x, we need to set the equation to zero. This is done by subtracting 15 from both sides:
x² - 4x - 12 = 0
Factoring the Quadratic Equation
The next step is to factor the quadratic equation. We need to find two numbers that add up to -4 and multiply to -12. These numbers are -6 and 2. Therefore, we can factor the equation as follows:
(x - 6)(x + 2) = 0
Solving for x
For the product of two factors to be zero, at least one of them must be zero. Therefore, we have two possible solutions:
- x - 6 = 0 => x = 6
- x + 2 = 0 => x = -2
Conclusion
The solutions to the quadratic equation (x-1)(x-3) = 15 are x = 6 and x = -2.