Question

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

Answer 1

What are the solutions to x2−121=0
? Enter the lesser number first.(1 point)
x=
, x=

Answer 2

To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we first rearrange the equation to bring all terms to one side:

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

Next, we want to express the left-hand side as a perfect square trinomial. To do this, we can complete the square.

1. Start with the coefficient of \( x \), which is \( 54 \). Take half of \( 54 \) and square it:
\[
\left(\frac{54}{2}\right)^2 = 27^2 = 729
\]

2. The expression \( x^2 + 54x + 729 \) can then be rewritten as:
\[
(x + 27)^2 = 729
\]

Next, we solve for \( x + 27 \):

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

Taking the square root of both sides gives:

\[
x + 27 = \pm 27
\]

Now we can solve for \( x \):

1. For the positive case:
\[
x + 27 = 27
\]
\[
x = 27 - 27 = 0
\]

2. For the negative case:
\[
x + 27 = -27
\]
\[
x = -27 - 27 = -54
\]

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

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