(2x+3)(3x+8/2-7x+1)=(x-5)(3x+8/2-7x+1)

3 min read Jun 16, 2024

Solving the Equation: (2x+3)(3x+8/2-7x+1) = (x-5)(3x+8/2-7x+1)

This equation presents a great opportunity to practice algebraic manipulation and equation solving. Let's break it down step-by-step:

1. Simplifying the Expression:

First, we need to simplify the expressions within the parentheses on both sides of the equation:

Left Side:
- (2x + 3)(3x + 4 - 7x + 1)
- (2x + 3)(-4x + 5)
Right Side:
- (x - 5)(3x + 4 - 7x + 1)
- (x - 5)(-4x + 5)

2. Expanding the Products:

Now, we need to expand the products on both sides using the distributive property (FOIL method):

Left Side:
- (2x + 3)(-4x + 5) = -8x² + 10x - 12x + 15 = -8x² - 2x + 15
Right Side:
- (x - 5)(-4x + 5) = -4x² + 5x + 20x - 25 = -4x² + 25x - 25

3. Combining Like Terms:

Next, combine the like terms on each side of the equation:

Left Side: -8x² - 2x + 15
Right Side: -4x² + 25x - 25

4. Rearranging the Equation:

To solve for x, we need to set the equation to zero. Let's move all terms to the left side:

-8x² - 2x + 15 - (-4x² + 25x - 25) = 0
-8x² - 2x + 15 + 4x² - 25x + 25 = 0
-4x² - 27x + 40 = 0

5. Solving the Quadratic Equation:

We now have a quadratic equation in standard form (ax² + bx + c = 0). To solve for x, we can use the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

In this case, a = -4, b = -27, and c = 40. Substitute these values into the formula and solve.

6. Finding the Solutions:

After plugging in the values and simplifying, you will get two solutions for x. These represent the values of x that satisfy the original equation.

Important Note: The quadratic formula can sometimes result in solutions that involve imaginary numbers.

By following these steps, you can successfully solve the equation (2x+3)(3x+8/2-7x+1) = (x-5)(3x+8/2-7x+1) and find the values of x that satisfy the equation.