Question
Solve \( (v+4)^{2}-24=0 \), where \( v \) is a real number. Simplify your answer as much as possible. If there is more than one solution, separate them with commas. If there is no solution, click "No solution." \( v=\square \)
Ask by Montgomery Haynes. in the United States
Jan 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( v = -2\sqrt{6} - 4, 2\sqrt{6} - 4 \)
Solution
Solve the equation \( (v+4)^{2}-24=0 \).
Solve the quadratic equation by following steps:
- step0: Solve using square roots:
\(\left(v+4\right)^{2}-24=0\)
- step1: Add or subtract both sides:
\(\left(v+4\right)^{2}=24\)
- step2: Simplify the expression:
\(v+4=\pm \sqrt{24}\)
- step3: Simplify the expression:
\(v+4=\pm 2\sqrt{6}\)
- step4: Separate into possible cases:
\(\begin{align}&v+4=2\sqrt{6}\\&v+4=-2\sqrt{6}\end{align}\)
- step5: Move the constant to the right side:
\(\begin{align}&v=2\sqrt{6}-4\\&v=-2\sqrt{6}-4\end{align}\)
- step6: Rewrite:
\(v_{1}=-2\sqrt{6}-4,v_{2}=2\sqrt{6}-4\)
The solutions to the equation \( (v+4)^{2}-24=0 \) are \( v=-2\sqrt{6}-4 \) and \( v=2\sqrt{6}-4 \).
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Beyond the Answer
To solve \( (v+4)^{2}-24=0 \), we start by isolating the square term: \[ (v+4)^{2} = 24 \] Next, we take the square root of both sides: \[ v+4 = \pm \sqrt{24} \] Since \( \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} \), we can rewrite the equation as: \[ v+4 = \pm 2\sqrt{6} \] Now, we solve for \( v \) by subtracting 4 from both sides: \[ v = -4 \pm 2\sqrt{6} \] This yields two solutions: \[ v = -4 + 2\sqrt{6}, \quad v = -4 - 2\sqrt{6} \] Thus, the solutions are: \[ v = -4 + 2\sqrt{6}, -4 - 2\sqrt{6} \]
