Question

Solve the following quadratic equation using the perfect square trinomial pattern: x2+54x=−729
.(1 point)

Answer 1

To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we can first rewrite the equation in a standard form.

1. Move the constant term to the left side:

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

2. Now, we want to complete the square for the left side \( x^2 + 54x + 729 \).

To complete the square:

- Take half of the coefficient of \( x \), which is \( 54 \), divide it by 2 to get \( 27 \).
- Square \( 27 \) to obtain \( 729 \).

This confirms we can rewrite the expression as a perfect square trinomial:

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

We can redefine this equation as:

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

3. Set the square equal to zero and solve for \( x \):

\[
x + 27 = 0
\]

\[
x = -27
\]

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

\[
\boxed{-27}
\]