Solve (x-4)² = (x+2)(x-1): Perfect Square Equals Product

$(x-4)^2=(x+2)(x-1)$

To solve the equation $(x-4)^2 = (x+2)(x-1)$, follow these detailed steps:

• Step 1: Expand the left side of the equation using the square of a binomial formula: $(x-4)^2 = x^2 - 8x + 16$.
• Step 2: Expand the right side using the distributive property: $(x+2)(x-1) = x(x-1) + 2(x-1) = x^2 - x + 2x - 2 = x^2 + x - 2$.
• Step 3: Set the expanded forms equal to each other: $x^2 - 8x + 16 = x^2 + x - 2$.
• Step 4: Subtract $x^2$ from both sides to simplify: $-8x + 16 = x - 2$.
• Step 5: Move all terms involving $x$ to one side and constant terms to the other: $-8x - x = -2 - 16$.
• Step 6: Combine like terms: $-9x = -18$.
• Step 7: Solve for $x$ by dividing both sides by $-9$: $x = 2$.

Therefore, the solution to the problem is $x = 2$.