Question
\( \quad(v-7)^{2}=2 v^{2}-8 v+54 \) If there is more than one solution, separate them with commas.
Ask by Griffiths Bolton. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are \( v = -1 \) and \( v = -5 \).
Solution
To solve the equation (v - 7)² = 2v² - 8v + 54, follow these steps:
1. Expand the left-hand side:
(v - 7)² = v² - 14v + 49
2. Substitute back into the equation:
v² - 14v + 49 = 2v² - 8v + 54
3. Bring all terms to one side:
0 = 2v² - 8v + 54 - (v² - 14v + 49)
0 = (2v² - v²) + (-8v + 14v) + (54 - 49)
0 = v² + 6v + 5
4. Factor the quadratic:
v² + 6v + 5 = (v + 1)(v + 5) = 0
5. Set each factor equal to zero:
v + 1 = 0 ⟹ v = -1
v + 5 = 0 ⟹ v = -5
Thus, the solutions are v = -1, -5.
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Simplify this solution The Deep Dive
To solve the equation \( (v-7)^{2} = 2v^{2} - 8v + 54 \), we first expand the left side: \[ (v - 7)^2 = v^2 - 14v + 49 \] Now, we can rewrite the equation as: \[ v^2 - 14v + 49 = 2v^2 - 8v + 54 \] Next, we will move all terms to one side: \[ 0 = 2v^2 - 8v + 54 - v^2 + 14v - 49 \] This simplifies to: \[ 0 = v^2 + 6v + 5 \] Now, we can factor the quadratic: \[ 0 = (v + 5)(v + 1) \] Setting each factor to zero gives us the solutions: \[ v + 5 = 0 \quad \Rightarrow \quad v = -5 \] \[ v + 1 = 0 \quad \Rightarrow \quad v = -1 \] Thus, the solutions to the equation are \( v = -5, -1 \).
