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, we first want to move all terms to one side of the equation:

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

Next, we can complete the square. The first step is to take the coefficient of \( x \), which is 54, divide it by 2, and then square the result:

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

Now, we can rewrite the quadratic equation as a perfect square trinomial:

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

Next, we take the square root of both sides:

\[
x + 27 = \pm \sqrt{729}.
\]

Since \( \sqrt{729} = 27 \), we have:

\[
x + 27 = 27 \quad \text{or} \quad x + 27 = -27.
\]

Now, we can solve for \( x \):

1. For \( x + 27 = 27 \):
\[
x = 27 - 27 = 0.
\]

2. For \( x + 27 = -27 \):
\[
x = -27 - 27 = -54.
\]

Thus, the solutions are:

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

So, the final answer is:

\[
x = 0, -54.
\]