(x%2B2)-(2x-3)(x%2B4)-x%2B14%3D0.jpeg "(x-1)(x+2)-(2x-3)(x+4)-x+14=0")
Solving the Equation: (x-1)(x+2)-(2x-3)(x+4)-x+14=0
This article will guide you through the steps involved in solving the equation (x-1)(x+2)-(2x-3)(x+4)-x+14=0.
1. Expand the products:
First, we need to expand the products using the distributive property (or FOIL method):
- (x-1)(x+2) = x² + 2x - x - 2 = x² + x - 2
- (2x-3)(x+4) = 2x² + 8x - 3x - 12 = 2x² + 5x - 12
Now, substitute these expanded expressions back into the original equation:
x² + x - 2 - (2x² + 5x - 12) - x + 14 = 0
2. Simplify the equation:
Simplify the equation by removing the parentheses and combining like terms:
x² + x - 2 - 2x² - 5x + 12 - x + 14 = 0 -x² - 5x + 24 = 0
3. Solve the quadratic equation:
We now have a quadratic equation in the form ax² + bx + c = 0. There are several ways to solve this:
-
Factoring: If the equation can be factored, this is the easiest method. However, in this case, factoring might be difficult.
-
Quadratic Formula: The quadratic formula is a reliable way to solve any quadratic equation. The formula is:
x = (-b ± √(b² - 4ac)) / 2a
In our equation, a = -1, b = -5, and c = 24. Plugging these values into the quadratic formula, we get:
x = (5 ± √((-5)² - 4 * -1 * 24)) / (2 * -1) x = (5 ± √(169)) / -2 x = (5 ± 13) / -2
-
Completing the Square: This method is also useful, but factoring or the quadratic formula are generally more efficient.
4. Find the solutions:
From the quadratic formula, we get two possible solutions:
- x = (5 + 13) / -2 = -9
- x = (5 - 13) / -2 = 4
Therefore, the solutions to the equation (x-1)(x+2)-(2x-3)(x+4)-x+14=0 are x = -9 and x = 4.