Question
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Solve this equation and select the choice that contains all of the solutions. \[ (x+7)^{2}=-16 \]

Ask by Stephens Warren. in the United States
Mar 14,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The solutions are \( x = -7 + 4i \) and \( x = -7 - 4i \).

Solution

We start with the equation: \[ (x+7)^2 = -16. \] 1. Take the square root of both sides: \[ x+7 = \pm \sqrt{-16}. \] 2. Recognize that \(\sqrt{-16} = 4i\) (where \(i\) is the imaginary unit): \[ x+7 = \pm 4i. \] 3. Solve for \(x\) by subtracting 7 from both sides: \[ x = -7 \pm 4i. \] Thus, the solutions are: \[ \{ -7+4i,\ -7-4i \}. \]

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The Deep Dive

First, we notice that the equation \((x+7)^{2}=-16\) presents a challenge because the square of any real number is non-negative, and thus it cannot equal a negative number like \(-16\). This means we will be working in the realm of complex numbers. To solve for \(x\), we start by taking the square root of both sides: \[ x + 7 = \pm \sqrt{-16}. \] The square root of \(-16\) can be simplified as: \[ \sqrt{-16} = 4i, \] where \(i\) is the imaginary unit. Thus, we have: \[ x + 7 = \pm 4i. \] To isolate \(x\), we can subtract 7 from both sides: \[ x = -7 \pm 4i. \] This gives us two solutions: \[ x = -7 + 4i \quad \text{and} \quad x = -7 - 4i. \] So the complete set of solutions is: \[ \{-7 + 4i, -7 - 4i\}. \]

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