Questions: Step 2: Factor the completed square Step 3: Use the Square Root Property Add -25 to both sides to get (x+4)^2=-25 Take the square root of both sides, remembering the square root property (there are two square roots to consider, the positive and the negative!) x+4=

Solution Steps

To solve the equation \((x+4)^2 = -25\), we need to take the square root of both sides. Since the right side is negative, the solutions will involve imaginary numbers. The square root property tells us that if \(a^2 = b\), then \(a = \pm \sqrt{b}\). Here, we apply this property to find the values of \(x\).

1. Recognize that \((x+4)^2 = -25\) implies \(x+4 = \pm \sqrt{-25}\).
2. Simplify \(\sqrt{-25}\) using imaginary numbers: \(\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i\).
3. Solve for \(x\) by isolating it on one side of the equation.

The given equation is already in the form of a completed square:

\[ (x+4)^2 = -25 \]

To solve for \(x\), we need to take the square root of both sides of the equation. Remember that when taking the square root of both sides, we consider both the positive and negative roots:

\[ x + 4 = \pm \sqrt{-25} \]

Since \(-25\) is a negative number, the square root will be an imaginary number. We can express \(\sqrt{-25}\) as \(5i\), where \(i\) is the imaginary unit (\(i = \sqrt{-1}\)).

Thus, we have:

\[ x + 4 = \pm 5i \]

To isolate \(x\), subtract 4 from both sides:

\[ x = -4 \pm 5i \]

This gives us two solutions:

1. \(x = -4 + 5i\)
2. \(x = -4 - 5i\)

Final Answer