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 move all terms to one side of the equation:

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

Next, we will complete the square. The first step in this process is to take half of the coefficient of \( x \) (which is 54), square it, and add it to both sides.

1. Half of 54 is \( \frac{54}{2} = 27 \).
2. Squaring 27 gives us \( 27^2 = 729 \).

Then we rewrite the equation incorporating this square:

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

This can be rewritten as:

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

Now we can take the square root of both sides:

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

Calculating the square root of 729 gives us:

\[
\sqrt{729} = 27
\]

So we have:

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

Now we solve for \( x \):

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

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

Thus, the solutions to the quadratic equation are:

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