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

Solve for (v).

[ 2 v^2+12 v+14=(v+3)^2 ]

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

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Solution Steps

Step 1: Set Up the Equation

We start with the equation: 2v2+12v+14=(v+3)2 2v^{2} + 12v + 14 = (v + 3)^{2}

Step 2: Expand the Right Side

Expand the right side of the equation: (v+3)2=v2+6v+9 (v + 3)^{2} = v^{2} + 6v + 9 Thus, the equation becomes: 2v2+12v+14=v2+6v+9 2v^{2} + 12v + 14 = v^{2} + 6v + 9

Step 3: Rearrange the Equation

Rearranging the equation to bring all terms to one side gives: 2v2+12v+14v26v9=0 2v^{2} + 12v + 14 - v^{2} - 6v - 9 = 0 This simplifies to: v2+6v+5=0 v^{2} + 6v + 5 = 0

Step 4: Factor the Quadratic

Next, we factor the quadratic equation: v2+6v+5=(v+5)(v+1)=0 v^{2} + 6v + 5 = (v + 5)(v + 1) = 0

Step 5: Solve for v v

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

Step 6: List the Solutions

The solutions for v v are: v=5,1 v = -5, -1

Final Answer

v=5,1 v = -5, -1

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