Question

Solve the following quadratic equation using the perfect square trinomial pattern: x^2+54x=−729

Answer 1

To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we first need to move all terms to one side of the equation to set it to zero:

\[
x^2 + 54x + 729 = 0
\]

Next, we will focus on the left side of the equation, \( x^2 + 54x \). To form a perfect square trinomial, we need to take half of the coefficient of \( x \), square it, and add it inside the expression.

1. Take half of the coefficient of \( x \), which is 54:
\[
\frac{54}{2} = 27
\]

2. Now square it:
\[
27^2 = 729
\]

Now we can rewrite the expression \( x^2 + 54x + 729 \) as a perfect square trinomial:

\[
x^2 + 54x + 729 = (x + 27)^2
\]

The equation now becomes:

\[
(x + 27)^2 = 0
\]

To find \( x \), we take the square root of both sides:

\[
x + 27 = 0
\]

Solving for \( x \):

\[
x = -27
\]

Thus, the solution to the equation \( x^2 + 54x = -729 \) is:

\[
\boxed{-27}
\]