Question

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

Answer 1

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

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

2. **Identify the coefficient of \( x \) to complete the square:**
The coefficient of \( x \) is 54. We take half of this coefficient and square it to find the number we need to complete the square.
\[
\left( \frac{54}{2} \right)^2 = 27^2 = 729
\]

3. **Rewrite the equation as a perfect square:**
Now, the expression \( x^2 + 54x + 729 \) can be rewritten as a perfect square trinomial:
\[
(x + 27)^2 = 0
\]

4. **Solve for \( x \):**
Set the equation to zero:
\[
(x + 27)^2 = 0
\]
Taking the square root of both sides, we find:
\[
x + 27 = 0
\]

5. **Isolate \( x \):**
\[
x = -27
\]

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