## Solving the Equation: (2x+1)(x-3) = x(4-x) - 9

This article will guide you through the steps involved in solving the equation **(2x+1)(x-3) = x(4-x) - 9**.

### Step 1: Expand Both Sides

First, we need to expand both sides of the equation to simplify it:

**Left Side:**(2x+1)(x-3) = 2x² - 6x + x - 3 = 2x² - 5x - 3**Right Side:**x(4-x) - 9 = 4x - x² - 9 = -x² + 4x - 9

Now, the equation becomes: **2x² - 5x - 3 = -x² + 4x - 9**

### Step 2: Combine Like Terms

To make the equation easier to solve, let's move all the terms to one side:

**Add x² to both sides:**3x² - 5x - 3 = 4x - 9**Subtract 4x from both sides:**3x² - 9x - 3 = -9**Add 9 to both sides:**3x² - 9x + 6 = 0

### Step 3: Solve the Quadratic Equation

We now have a quadratic equation in the standard form (ax² + bx + c = 0). We can solve this using the quadratic formula:

**Quadratic Formula:**x = (-b ± √(b² - 4ac)) / 2a**In our equation:**a = 3, b = -9, c = 6

Plugging these values into the quadratic formula:

x = (9 ± √((-9)² - 4 * 3 * 6)) / (2 * 3) x = (9 ± √(81 - 72)) / 6 x = (9 ± √9) / 6 x = (9 ± 3) / 6

This gives us two possible solutions:

**x1 = (9 + 3) / 6 = 2****x2 = (9 - 3) / 6 = 1**

### Conclusion

Therefore, the solutions to the equation (2x+1)(x-3) = x(4-x) - 9 are **x = 2** and **x = 1**.