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Solving the Quadratic Equation: (x+4)(x-9) = 4x
This article will guide you through solving the quadratic equation (x+4)(x-9) = 4x. We will use algebraic manipulation to arrive at the solution(s) for x.
Expanding and Simplifying the Equation
First, we need to expand the left side of the equation by multiplying the terms:
(x + 4)(x - 9) = x² - 5x - 36
Now, let's move the 4x term from the right side to the left side by subtracting it from both sides:
x² - 5x - 36 - 4x = 0
Combining like terms, we get the simplified quadratic equation:
x² - 9x - 36 = 0
Solving the Quadratic Equation
We can solve this equation using a few different methods, including factoring, completing the square, or the quadratic formula. In this case, factoring is the most straightforward method.
Factoring the Equation
We need to find two numbers that add up to -9 (the coefficient of the x term) and multiply to -36 (the constant term). These numbers are -12 and 3:
(x - 12)(x + 3) = 0
For the product of two factors to equal zero, one or both of the factors must be equal to zero. Therefore, we have two possible solutions:
x - 12 = 0 or x + 3 = 0
Solving for x in each case:
x = 12 or x = -3
Conclusion
Therefore, the solutions to the quadratic equation (x+4)(x-9) = 4x are x = 12 and x = -3.