Solve the following quadratic equation using the perfect square trinomial pattern: x2+54x=−729

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1 answer

To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we first rearrange the equation to set it to zero:

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

Next, we want to use the method of completing the square. To complete the square for the expression \( x^2 + 54x \), we follow these steps:

  1. Take the coefficient of \( x \), which is 54, divide it by 2, and then square it:

\[ \left(\frac{54}{2}\right)^2 = 27^2 = 729 \]

  1. We can express \( x^2 + 54x + 729 \) as a perfect square:

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

This step gives us:

\[ (x + 27)^2 - 729 = 0 \]

Alternatively, we can write it as:

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

  1. Next, we take the square root of both sides:

\[ x + 27 = \pm \sqrt{729} \]

Since \( \sqrt{729} = 27 \), we have:

\[ x + 27 = 27 \quad \text{or} \quad x + 27 = -27 \]

  1. Solving each equation gives us:

For \( x + 27 = 27 \):

\[ x = 27 - 27 = 0 \]

For \( x + 27 = -27 \):

\[ x = -27 - 27 = -54 \]

Thus, the solutions to the equation \( x^2 + 54x + 729 = 0 \) are:

\[ \boxed{0 \text{ and } -54} \]