(x+4)^2=16x

3 min read Jun 16, 2024

Solving the Equation (x+4)^2 = 16x

This article will guide you through the process of solving the equation (x+4)^2 = 16x. We will use algebraic techniques to find the solutions for x.

1. Expanding the Equation

First, we need to expand the left side of the equation by using the FOIL (First, Outer, Inner, Last) method:

(x+4)^2 = (x+4)(x+4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16

Now our equation becomes:

x^2 + 8x + 16 = 16x

2. Rearranging the Equation

To solve for x, we need to rearrange the equation to have all terms on one side:

x^2 + 8x + 16 - 16x = 0

This simplifies to:

x^2 - 8x + 16 = 0

3. Solving the Quadratic Equation

The equation we have now is a quadratic equation. There are two main ways to solve quadratic equations:

Factoring: In this case, the equation can be factored easily: (x - 4)(x - 4) = 0 This gives us the solution x = 4. Since the factor (x - 4) is repeated, we have a double root, meaning the solution x = 4 appears twice.
Quadratic Formula: The quadratic formula can be used to solve any quadratic equation in the form ax^2 + bx + c = 0. The formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

In our equation, a = 1, b = -8, and c = 16. Plugging these values into the formula gives:

x = (8 ± √((-8)^2 - 4 * 1 * 16)) / (2 * 1) x = (8 ± √(0)) / 2 x = 4

Conclusion

The solution to the equation (x+4)^2 = 16x is x = 4. We found this solution using both factoring and the quadratic formula, confirming the solution.