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6v^{2}+13v-8+3=0
Add 3 to both sides.
6v^{2}+13v-5=0
Add -8 and 3 to get -5.
a+b=13 ab=6\left(-5\right)=-30
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6v^{2}+av+bv-5. To find a and b, set up a system to be solved.
-1,30 -2,15 -3,10 -5,6
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 -30.
-1+30=29 -2+15=13 -3+10=7 -5+6=1
Calculate the sum for each pair.
a=-2 b=15
The solution is the pair that gives sum 13.
\left(6v^{2}-2v\right)+\left(15v-5\right)
Rewrite 6v^{2}+13v-5 as \left(6v^{2}-2v\right)+\left(15v-5\right).
2v\left(3v-1\right)+5\left(3v-1\right)
Factor out 2v in the first and 5 in the second group.
\left(3v-1\right)\left(2v+5\right)
Factor out common term 3v-1 by using distributive property.
v=\frac{1}{3} v=-\frac{5}{2}
To find equation solutions, solve 3v-1=0 and 2v+5=0.
6v^{2}+13v-8=-3
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.
6v^{2}+13v-8-\left(-3\right)=-3-\left(-3\right)
Add 3 to both sides of the equation.
6v^{2}+13v-8-\left(-3\right)=0
Subtracting -3 from itself leaves 0.
6v^{2}+13v-5=0
Subtract -3 from -8.
v=\frac{-13±\sqrt{13^{2}-4\times 6\left(-5\right)}}{2\times 6}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 13 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
v=\frac{-13±\sqrt{169-4\times 6\left(-5\right)}}{2\times 6}
Square 13.
v=\frac{-13±\sqrt{169-24\left(-5\right)}}{2\times 6}
Multiply -4 times 6.
v=\frac{-13±\sqrt{169+120}}{2\times 6}
Multiply -24 times -5.
v=\frac{-13±\sqrt{289}}{2\times 6}
Add 169 to 120.
v=\frac{-13±17}{2\times 6}
Take the square root of 289.
v=\frac{-13±17}{12}
Multiply 2 times 6.
v=\frac{4}{12}
Now solve the equation v=\frac{-13±17}{12} when ± is plus. Add -13 to 17.
v=\frac{1}{3}
Reduce the fraction \frac{4}{12} to lowest terms by extracting and canceling out 4.
v=-\frac{30}{12}
Now solve the equation v=\frac{-13±17}{12} when ± is minus. Subtract 17 from -13.
v=-\frac{5}{2}
Reduce the fraction \frac{-30}{12} to lowest terms by extracting and canceling out 6.
v=\frac{1}{3} v=-\frac{5}{2}
The equation is now solved.
6v^{2}+13v-8=-3
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.
6v^{2}+13v-8-\left(-8\right)=-3-\left(-8\right)
Add 8 to both sides of the equation.
6v^{2}+13v=-3-\left(-8\right)
Subtracting -8 from itself leaves 0.
6v^{2}+13v=5
Subtract -8 from -3.
\frac{6v^{2}+13v}{6}=\frac{5}{6}
Divide both sides by 6.
v^{2}+\frac{13}{6}v=\frac{5}{6}
Dividing by 6 undoes the multiplication by 6.
v^{2}+\frac{13}{6}v+\left(\frac{13}{12}\right)^{2}=\frac{5}{6}+\left(\frac{13}{12}\right)^{2}
Divide \frac{13}{6}, the coefficient of the x term, by 2 to get \frac{13}{12}. Then add the square of \frac{13}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
v^{2}+\frac{13}{6}v+\frac{169}{144}=\frac{5}{6}+\frac{169}{144}
Square \frac{13}{12} by squaring both the numerator and the denominator of the fraction.
v^{2}+\frac{13}{6}v+\frac{169}{144}=\frac{289}{144}
Add \frac{5}{6} to \frac{169}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(v+\frac{13}{12}\right)^{2}=\frac{289}{144}
Factor v^{2}+\frac{13}{6}v+\frac{169}{144}. 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+\frac{13}{12}\right)^{2}}=\sqrt{\frac{289}{144}}
Take the square root of both sides of the equation.
v+\frac{13}{12}=\frac{17}{12} v+\frac{13}{12}=-\frac{17}{12}
Simplify.
v=\frac{1}{3} v=-\frac{5}{2}
Subtract \frac{13}{12} from both sides of the equation.