Questions: Solve for (v). [ 2 v^2+12 v+14=(v+3)^2 ] [ v= ]

Transcript text: Solve for $v$. \[ 2 v^{2}+12 v+14=(v+3)^{2} \] \[ v= \]

Solution

Solution Steps

Step 1: Set Up the Equation

We start with the equation: \[ 2v^{2} + 12v + 14 = (v + 3)^{2} \]

Step 2: Expand the Right Side

Expand the right side of the equation: \[ (v + 3)^{2} = v^{2} + 6v + 9 \] Thus, the equation becomes: \[ 2v^{2} + 12v + 14 = v^{2} + 6v + 9 \]

Step 3: Rearrange the Equation

Rearranging the equation to bring all terms to one side gives: \[ 2v^{2} + 12v + 14 - v^{2} - 6v - 9 = 0 \] This simplifies to: \[ v^{2} + 6v + 5 = 0 \]

Step 4: Factor the Quadratic

Next, we factor the quadratic equation: \[ v^{2} + 6v + 5 = (v + 5)(v + 1) = 0 \]

Step 5: Solve for $ v $

Setting each factor equal to zero gives the solutions: \[ v + 5 = 0 \quad \Rightarrow \quad v = -5 \] \[ v + 1 = 0 \quad \Rightarrow \quad v = -1 \]

Step 6: List the Solutions

The solutions for $ v $ are: \[ v = -5, -1 \]

Final Answer

$ v = -5, -1 $

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