Question

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

Answer 1

To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we first rearrange the equation to set it to zero:

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

Next, we want to use the method of completing the square. To complete the square for the expression \( x^2 + 54x \), we follow these steps:

1. Take the coefficient of \( x \), which is 54, divide it by 2, and then square it:

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

2. We can express \( x^2 + 54x + 729 \) as a perfect square:

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

This step gives us:

\[
(x + 27)^2 - 729 = 0
\]

Alternatively, we can write it as:

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

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

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

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

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

4. Solving each equation gives us:

For \( x + 27 = 27 \):

\[
x = 27 - 27 = 0
\]

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

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

Thus, the solutions to the equation \( x^2 + 54x + 729 = 0 \) are:

\[
\boxed{0 \text{ and } -54}
\]