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Solving the Equation: (1/2x-3)(x+2) = (5-x)(6-x)
This article will guide you through the process of solving the equation (1/2x-3)(x+2) = (5-x)(6-x).
Expanding the Equation
First, we need to expand both sides of the equation using the distributive property (also known as FOIL):
- Left Side: (1/2x - 3)(x + 2) = (1/2x * x) + (1/2x * 2) + (-3 * x) + (-3 * 2) = (1/2)x² - x - 6
- Right Side: (5 - x)(6 - x) = (5 * 6) + (5 * -x) + (-x * 6) + (-x * -x) = x² - 11x + 30
Now our equation becomes: (1/2)x² - x - 6 = x² - 11x + 30
Bringing Terms Together
To solve for x, we need to bring all terms to one side of the equation. Let's move all the terms to the left side:
(1/2)x² - x - 6 - x² + 11x - 30 = 0
Combining like terms:
(-1/2)x² + 10x - 36 = 0
Solving for x
To solve for x, we can use several methods, including:
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Quadratic Formula: This is a general formula that can be used to solve any quadratic equation. The quadratic formula is: x = [-b ± √(b² - 4ac)] / 2a, where a, b, and c are the coefficients of the quadratic equation. In our equation, a = -1/2, b = 10, and c = -36. Substituting these values into the quadratic formula will give us two solutions for x.
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Factoring: If the quadratic equation can be factored, this can be a simpler method. However, in this case, factoring might be challenging.
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Completing the Square: This method involves manipulating the equation to form a perfect square trinomial. While it can be a useful method, it's often more complex than using the quadratic formula.
Finding the Solutions
After applying one of the methods above, you will obtain two solutions for x. These solutions represent the values of x that satisfy the original equation.
It's important to remember that not all equations have real solutions. In this case, it is possible that the equation has no real solutions.
Checking the Solutions
After finding the solutions, you should always check them by substituting them back into the original equation to ensure they are valid.
By following these steps, you can successfully solve the equation (1/2x-3)(x+2) = (5-x)(6-x) and find the values of x that satisfy the equation.