## Solving the Equation: (x+2)(x-3)-(2x-5)(x+3)=x(x-5)

This article will guide you through the steps of solving the given equation:

**(x+2)(x-3)-(2x-5)(x+3)=x(x-5)**

### Expanding the Equation

To begin, we need to expand the products on both sides of the equation:

**Left Side:**- (x+2)(x-3) = x² - x - 6
- (2x-5)(x+3) = 2x² + x - 15

**Right Side:**- x(x-5) = x² - 5x

Now the equation becomes:

**x² - x - 6 - (2x² + x - 15) = x² - 5x**

### Simplifying the Equation

Next, we simplify the left side by distributing the negative sign:

**x² - x - 6 - 2x² - x + 15 = x² - 5x**

Combining like terms on the left side gives:

**-x² - 2x + 9 = x² - 5x**

### Rearranging and Solving

To solve for x, we need to bring all the terms to one side. Let's move all terms to the left side:

**-x² - 2x + 9 - x² + 5x = 0**

Combining like terms:

**-2x² + 3x + 9 = 0**

Now we have a quadratic equation. We can solve this using the quadratic formula:

**x = (-b ± √(b² - 4ac)) / 2a**

Where:

- a = -2
- b = 3
- c = 9

Substituting these values into the quadratic formula:

**x = (-3 ± √(3² - 4 * -2 * 9)) / (2 * -2)**

**x = (-3 ± √(9 + 72)) / -4**

**x = (-3 ± √81) / -4**

**x = (-3 ± 9) / -4**

This gives us two possible solutions:

**x = (-3 + 9) / -4 = -1.5****x = (-3 - 9) / -4 = 3**

### Conclusion

Therefore, the solutions to the equation (x+2)(x-3)-(2x-5)(x+3)=x(x-5) are **x = -1.5** and **x = 3**.