(x+1)^2=9x+19

2 min read Jun 16, 2024

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.