Solve 3v^2=11v+4

$$3 v ^ { 2 } = 11 v + 4$$

Solve for v

$$v=-\frac{1}{3}\approx -0.333333333$$

$$v=4$$

Steps Using Factoring By Grouping

Subtract 11v from both sides.

$$3v^{2}-11v=4$$

Subtract 4 from both sides.

$$3v^{2}-11v-4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3v^{2}+av+bv-4. To find a and b, set up a system to be solved.

$$a+b=-11 ab=3\left(-4\right)=-12$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -12.

$$1,-12 2,-6 3,-4$$

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -11.

$$a=-12 b=1$$

Rewrite 3v^{2}-11v-4 as \left(3v^{2}-12v\right)+\left(v-4\right).

$$\left(3v^{2}-12v\right)+\left(v-4\right)$$

Factor out 3v in 3v^{2}-12v.

$$3v\left(v-4\right)+v-4$$

Factor out common term v-4 by using distributive property.

$$\left(v-4\right)\left(3v+1\right)$$

To find equation solutions, solve v-4=0 and 3v+1=0.

$$v=4 v=-\frac{1}{3}$$

Steps Using the Quadratic Formula

Subtract 11v from both sides.

$$3v^{2}-11v=4$$

Subtract 4 from both sides.

$$3v^{2}-11v-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -11 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 3\left(-4\right)}}{2\times 3}$$

Square -11.

$$v=\frac{-\left(-11\right)±\sqrt{121-4\times 3\left(-4\right)}}{2\times 3}$$

Multiply -4 times 3.

$$v=\frac{-\left(-11\right)±\sqrt{121-12\left(-4\right)}}{2\times 3}$$

Multiply -12 times -4.

$$v=\frac{-\left(-11\right)±\sqrt{121+48}}{2\times 3}$$

Add 121 to 48.

$$v=\frac{-\left(-11\right)±\sqrt{169}}{2\times 3}$$

Take the square root of 169.

$$v=\frac{-\left(-11\right)±13}{2\times 3}$$

The opposite of -11 is 11.

$$v=\frac{11±13}{2\times 3}$$

Multiply 2 times 3.

$$v=\frac{11±13}{6}$$

Now solve the equation v=\frac{11±13}{6} when ± is plus. Add 11 to 13.

$$v=\frac{24}{6}$$

Divide 24 by 6.

$$v=4$$

Now solve the equation v=\frac{11±13}{6} when ± is minus. Subtract 13 from 11.

$$v=-\frac{2}{6}$$

Reduce the fraction \frac{-2}{6} to lowest terms by extracting and canceling out 2.

$$v=-\frac{1}{3}$$

The equation is now solved.

$$v=4 v=-\frac{1}{3}$$

Steps for Completing the Square

Subtract 11v from both sides.

$$3v^{2}-11v=4$$

Divide both sides by 3.

$$\frac{3v^{2}-11v}{3}=\frac{4}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$v^{2}-\frac{11}{3}v=\frac{4}{3}$$

Divide -\frac{11}{3}, the coefficient of the x term, by 2 to get -\frac{11}{6}. Then add the square of -\frac{11}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-\frac{11}{3}v+\left(-\frac{11}{6}\right)^{2}=\frac{4}{3}+\left(-\frac{11}{6}\right)^{2}$$

Square -\frac{11}{6} by squaring both the numerator and the denominator of the fraction.

$$v^{2}-\frac{11}{3}v+\frac{121}{36}=\frac{4}{3}+\frac{121}{36}$$

Add \frac{4}{3} to \frac{121}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$v^{2}-\frac{11}{3}v+\frac{121}{36}=\frac{169}{36}$$

Factor v^{2}-\frac{11}{3}v+\frac{121}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v-\frac{11}{6}\right)^{2}=\frac{169}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-\frac{11}{6}\right)^{2}}=\sqrt{\frac{169}{36}}$$

Simplify.

$$v-\frac{11}{6}=\frac{13}{6} v-\frac{11}{6}=-\frac{13}{6}$$

Add \frac{11}{6} to both sides of the equation.

$$v=4 v=-\frac{1}{3}$$