Question

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

Answer 1

To solve the equation \(x^2 + 54x = -729\) using the perfect square trinomial pattern, we start by moving \(-729\) to the left side of the equation:

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

Next, we aim to rewrite the left side as a perfect square trinomial. The perfect square trinomial is expressed as \((x + a)^2\). To find \(a\), we take the coefficient of \(x\), which is \(54\), divide it by \(2\), and then square it:

\[
a = \frac{54}{2} = 27
\]

Then we compute \(a^2\):

\[
27^2 = 729
\]

Thus, we can write the left-hand side of the equation as:

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

Next, we take the square root of both sides:

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

Calculating the square root gives:

\[
\sqrt{729} = 27
\]

So we have two cases to consider:

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

Now we solve each case:

1. For \(x + 27 = 27\):

\[
x = 27 - 27 = 0
\]

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

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

Thus, the solutions to the equation are:

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

So, the final answer is:

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