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Solving the Quadratic Equation: (x-2)(x+4) + 9 = 0
This article will guide you through the process of solving the quadratic equation (x-2)(x+4) + 9 = 0.
Expanding and Simplifying the Equation
First, we need to expand the equation by multiplying the terms in the parentheses:
(x-2)(x+4) + 9 = 0
x² + 4x - 2x - 8 + 9 = 0
Combining like terms, we get:
x² + 2x + 1 = 0
Solving by Factoring
This equation can be factored into:
(x+1)(x+1) = 0
Therefore, the solution is:
x = -1
Solving by the Quadratic Formula
The quadratic formula is a general solution for equations in the form ax² + bx + c = 0. In this case, a = 1, b = 2, and c = 1.
The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a
Substituting the values:
x = (-2 ± √(2² - 4 * 1 * 1)) / (2 * 1)
x = (-2 ± √0) / 2
x = -1
Conclusion
Both methods, factoring and using the quadratic formula, arrive at the same solution: x = -1.
It's important to remember that quadratic equations can have up to two solutions. In this particular case, we have a double root, meaning the solution x = -1 appears twice.