Question
Solve by completing the square. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.) \[ v^{2}+6 v+1=0 \] \( v=\square \)
Ask by Harmon Wyatt. in the United States
Nov 21,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( v=-3-2\sqrt{2}, -3+2\sqrt{2} \)
Solution
Solve the equation \( v^{2}+6v+1=0 \).
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
\(v^{2}+6v+1=0\)
- step1: Solve using the quadratic formula:
\(v=\frac{-6\pm \sqrt{6^{2}-4}}{2}\)
- step2: Simplify the expression:
\(v=\frac{-6\pm \sqrt{32}}{2}\)
- step3: Simplify the expression:
\(v=\frac{-6\pm 4\sqrt{2}}{2}\)
- step4: Separate into possible cases:
\(\begin{align}&v=\frac{-6+4\sqrt{2}}{2}\\&v=\frac{-6-4\sqrt{2}}{2}\end{align}\)
- step5: Simplify the expression:
\(\begin{align}&v=-3+2\sqrt{2}\\&v=\frac{-6-4\sqrt{2}}{2}\end{align}\)
- step6: Simplify the expression:
\(\begin{align}&v=-3+2\sqrt{2}\\&v=-3-2\sqrt{2}\end{align}\)
- step7: Rewrite:
\(v_{1}=-3-2\sqrt{2},v_{2}=-3+2\sqrt{2}\)
The solutions to the equation \(v^{2}+6v+1=0\) by completing the square are \(v=-3-2\sqrt{2}\) and \(v=-3+2\sqrt{2}\).
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Simplify this solution Mind Expander
To solve the equation \( v^{2}+6v+1=0 \) by completing the square, we first move the constant term to the other side: \( v^2 + 6v = -1 \). Next, we take half of the coefficient of \( v \), square it, and add it to both sides: Half of 6 is 3, squaring it gives us 9. So we add 9 to both sides: \( v^2 + 6v + 9 = 8 \). Now, we can express the left side as a perfect square: \( (v + 3)^2 = 8 \). Taking the square root of both sides gives: \( v + 3 = \pm\sqrt{8} = \pm 2\sqrt{2} \). Subtracting 3 from both sides leads to our two solutions: \( v = -3 + 2\sqrt{2}, -3 - 2\sqrt{2} \). The solutions are: \( v = -3 + 2\sqrt{2}, -3 - 2\sqrt{2} \).
