Solve the following quadratic equation using the perfect square trinomial pattern: x^2+54x=−729

Question

.(1 point)
x=___

Answer 1

To solve the 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 want to complete the square. For the quadratic expression \( x^2 + 54x \), we take half of the coefficient of \( x \) (which is 54), square it, and add it to both sides of the equation.

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

Thus, we can rewrite the equation as:

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

Now we can solve for \( x \):

1. Take the square root of both sides:

\[ x + 27 = 0. \]

2. Subtract 27 from both sides:

\[ x = -27. \]

So the solution to the equation is:

\[ x = -27. \]

The final answer is \( \boxed{-27} \).