Question

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

Answer 1

To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, follow these steps:

1. Move all terms to one side of the equation:
\[
x^2 + 54x + 729 = 0
\]

2. Rewrite the left side as a perfect square trinomial. To do this, we need to complete the square for \( x^2 + 54x \):
- Take half of the coefficient of \( x \) (which is 54), square it, and add it to both sides of the equation:
\[
\left(\frac{54}{2}\right)^2 = 27^2 = 729
\]

3. Now, rewrite the equation:
\[
(x + 27)^2 = 0
\]

4. Take the square root of both sides:
\[
x + 27 = 0
\]

5. Solve for \( x \):
\[
x = -27
\]

Thus, the solution to the equation \( x^2 + 54x = -729 \) is:
\[
x = -27
\]