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 move all terms to one side of the equation:

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

Next, we will complete the square. The first step in this process is to take half of the coefficient of \( x \) (which is 54), square it, and add it to both sides.

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

Then we rewrite the equation incorporating this square:

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

This can be rewritten as:

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

Now we can take the square root of both sides:

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

Calculating the square root of 729 gives us:

\[ \sqrt{729} = 27 \]

So we have:

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

Now we solve for \( x \):

  1. For \( x + 27 = 27 \): \[ x = 27 - 27 = 0 \]

  2. For \( x + 27 = -27 \): \[ x = -27 - 27 = -54 \]

Thus, the solutions to the quadratic equation are:

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