We start with the equation: 2v2+12v+14=(v+3)2 2v^{2} + 12v + 14 = (v + 3)^{2} 2v2+12v+14=(v+3)2
Expand the right side of the equation: (v+3)2=v2+6v+9 (v + 3)^{2} = v^{2} + 6v + 9 (v+3)2=v2+6v+9 Thus, the equation becomes: 2v2+12v+14=v2+6v+9 2v^{2} + 12v + 14 = v^{2} + 6v + 9 2v2+12v+14=v2+6v+9
Rearranging the equation to bring all terms to one side gives: 2v2+12v+14−v2−6v−9=0 2v^{2} + 12v + 14 - v^{2} - 6v - 9 = 0 2v2+12v+14−v2−6v−9=0 This simplifies to: v2+6v+5=0 v^{2} + 6v + 5 = 0 v2+6v+5=0
Next, we factor the quadratic equation: v2+6v+5=(v+5)(v+1)=0 v^{2} + 6v + 5 = (v + 5)(v + 1) = 0 v2+6v+5=(v+5)(v+1)=0
Setting each factor equal to zero gives the solutions: v+5=0⇒v=−5 v + 5 = 0 \quad \Rightarrow \quad v = -5 v+5=0⇒v=−5 v+1=0⇒v=−1 v + 1 = 0 \quad \Rightarrow \quad v = -1 v+1=0⇒v=−1
The solutions for v v v are: v=−5,−1 v = -5, -1 v=−5,−1
v=−5,−1 v = -5, -1 v=−5,−1
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