Solving Quadratic Equations Using Patterns Practice

Question

Complete this assessment to review what you’ve learned. It will not count toward your grade.
4 of 54 of 5 Items

Question
Solve the following quadratic equation using the perfect square trinomial pattern: x2+54x=−729
.(1 point)
x=
Remaining Attempts : 3
Skip to navigation

Answer 1

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

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

Next, we will complete the square. The goal is to rewrite the left side as a perfect square trinomial. A perfect square trinomial has the form \( (x + a)^2 \).

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

2. Therefore, we can rewrite the left side: \[ x^2 + 54x + 729 = (x + 27)^2 \]

Now we set the equation equal to zero:

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

3. To solve for \( x \), take the square root of both sides: \[ x + 27 = 0 \]

4. Finally, solve for \( x \): \[ x = -27 \]

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

\[ \boxed{-27} \]