Solve the Equation: (x-4)² - x(x+8) = 16 Step by Step

To solve this problem, we'll perform the following steps:

• Step 1: Expand $(x-4)^2$
• Step 2: Expand and simplify $-x(x+8)$
• Step 3: Combine like terms to form a quadratic equation
• Step 4: Solve the quadratic equation by factoring

Let's go through these steps with detailed explanations:

Step 1: Expand $(x-4)^2$.

Using the formula $(a-b)^2 = a^2 - 2ab + b^2$, we get:

$(x-4)^2 = x^2 - 8x + 16$.

Step 2: Expand $-x(x+8)$.

Using the distributive property, $-x(x+8) = -x^2 - 8x$.

Step 3: Combine both parts and simplify:

We have:

$x^2 - 8x + 16 - x^2 - 8x = 16$.

Simplify by combining like terms:

-16x + 16 = 16.

Subtract 16 from both sides:

-16x = 0.

Solve for $x$:

-16x = 0 \implies x = 0.

Thus, the solution to the equation is $x = 0$.