Question

Solve the following quadratic equation using the perfect square trinomial pattern: x^2+54x=−729 .(1 point)
x=

Answer 1

To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, first, we need to move the constant term to the other side:

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

Next, we can rewrite the left side as a perfect square trinomial. We look for a number that when squared gives us the constant term, which is 729 when we complete the square.

1. First, take half of the \( b \) term (which is 54), square it, and check:

\[
\left(\frac{54}{2}\right)^2 = 27^2 = 729
\]

2. Now, we can write the left side as a square:

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

3. Next, we take the square root of both sides:

\[
x + 27 = \pm 27
\]

This gives us two equations to solve:

1. \( x + 27 = 27 \)
2. \( x + 27 = -27 \)

**Solving the first equation:**

\[
x + 27 = 27 \\
x = 27 - 27 \\
x = 0
\]

**Solving the second equation:**

\[
x + 27 = -27 \\
x = -27 - 27 \\
x = -54
\]

Thus, the solutions for the equation \( x^2 + 54x = -729 \) are:

\[
x = 0 \quad \text{and} \quad x = -54
\]

Final answer:

\[
x = 0 \text{ or } x = -54
\]