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3v^{2}+6v-45=0
Subtract 45 from both sides.
v^{2}+2v-15=0
Divide both sides by 3.
a+b=2 ab=1\left(-15\right)=-15
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as v^{2}+av+bv-15. To find a and b, set up a system to be solved.
-1,15 -3,5
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.
-1+15=14 -3+5=2
Calculate the sum for each pair.
a=-3 b=5
The solution is the pair that gives sum 2.
\left(v^{2}-3v\right)+\left(5v-15\right)
Rewrite v^{2}+2v-15 as \left(v^{2}-3v\right)+\left(5v-15\right).
v\left(v-3\right)+5\left(v-3\right)
Factor out v in the first and 5 in the second group.
\left(v-3\right)\left(v+5\right)
Factor out common term v-3 by using distributive property.
v=3 v=-5
To find equation solutions, solve v-3=0 and v+5=0.
3v^{2}+6v=45
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
3v^{2}+6v-45=45-45
Subtract 45 from both sides of the equation.
3v^{2}+6v-45=0
Subtracting 45 from itself leaves 0.
v=\frac{-6±\sqrt{6^{2}-4\times 3\left(-45\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 6 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
v=\frac{-6±\sqrt{36-4\times 3\left(-45\right)}}{2\times 3}
Square 6.
v=\frac{-6±\sqrt{36-12\left(-45\right)}}{2\times 3}
Multiply -4 times 3.
v=\frac{-6±\sqrt{36+540}}{2\times 3}
Multiply -12 times -45.
v=\frac{-6±\sqrt{576}}{2\times 3}
Add 36 to 540.
v=\frac{-6±24}{2\times 3}
Take the square root of 576.
v=\frac{-6±24}{6}
Multiply 2 times 3.
v=\frac{18}{6}
Now solve the equation v=\frac{-6±24}{6} when ± is plus. Add -6 to 24.
v=3
Divide 18 by 6.
v=-\frac{30}{6}
Now solve the equation v=\frac{-6±24}{6} when ± is minus. Subtract 24 from -6.
v=-5
Divide -30 by 6.
v=3 v=-5
The equation is now solved.
3v^{2}+6v=45
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{3v^{2}+6v}{3}=\frac{45}{3}
Divide both sides by 3.
v^{2}+\frac{6}{3}v=\frac{45}{3}
Dividing by 3 undoes the multiplication by 3.
v^{2}+2v=\frac{45}{3}
Divide 6 by 3.
v^{2}+2v=15
Divide 45 by 3.
v^{2}+2v+1^{2}=15+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
v^{2}+2v+1=15+1
Square 1.
v^{2}+2v+1=16
Add 15 to 1.
\left(v+1\right)^{2}=16
Factor v^{2}+2v+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(v+1\right)^{2}}=\sqrt{16}
Take the square root of both sides of the equation.
v+1=4 v+1=-4
Simplify.
v=3 v=-5
Subtract 1 from both sides of the equation.