## Solving the Quadratic Equation: (x+1)^2 = 9x + 19

This article will guide you through the process of solving the quadratic equation **(x+1)^2 = 9x + 19**. We will use algebraic manipulation to simplify the equation and then apply the quadratic formula to find the solutions.

### Expanding and Rearranging the Equation

**Expand the left side:**(x + 1)^2 = (x + 1)(x + 1) = x^2 + 2x + 1**Subtract 9x and 19 from both sides:**x^2 + 2x + 1 - 9x - 19 = 0**Simplify:**x^2 - 7x - 18 = 0

### Applying the Quadratic Formula

Now that we have a standard quadratic equation in the form ax^2 + bx + c = 0, we can use the quadratic formula to solve for x:

**x = (-b ± √(b^2 - 4ac)) / 2a**

Where:

- a = 1
- b = -7
- c = -18

Substituting these values into the quadratic formula:

x = (7 ± √((-7)^2 - 4 * 1 * -18)) / (2 * 1) x = (7 ± √(49 + 72)) / 2 x = (7 ± √121) / 2 x = (7 ± 11) / 2

### Finding the Solutions

This gives us two possible solutions:

**x = (7 + 11) / 2 = 9****x = (7 - 11) / 2 = -2**

### Conclusion

Therefore, the solutions to the quadratic equation (x+1)^2 = 9x + 19 are **x = 9** and **x = -2**.