Solve 3v^2+3=-10v

$$3 v ^ { 2 } + 3 = - 10 v$$

Solve for v

$$v=-3$$

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

Steps Using Factoring By Grouping

Add 10v to both sides.

$$3v^{2}+3+10v=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$3v^{2}+10v+3=0$$

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

$$a+b=10 ab=3\times 3=9$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 9.

$$1,9 3,3$$

Calculate the sum for each pair.

$$1+9=10 3+3=6$$

The solution is the pair that gives sum 10.

$$a=1 b=9$$

Rewrite 3v^{2}+10v+3 as \left(3v^{2}+v\right)+\left(9v+3\right).

$$\left(3v^{2}+v\right)+\left(9v+3\right)$$

Factor out v in the first and 3 in the second group.

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

Factor out common term 3v+1 by using distributive property.

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

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

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

Steps Using the Quadratic Formula

Add 10v to both sides.

$$3v^{2}+3+10v=0$$

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.

$$3v^{2}+10v+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 10 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-10±\sqrt{10^{2}-4\times 3\times 3}}{2\times 3}$$

Square 10.

$$v=\frac{-10±\sqrt{100-4\times 3\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$v=\frac{-10±\sqrt{100-12\times 3}}{2\times 3}$$

Multiply -12 times 3.

$$v=\frac{-10±\sqrt{100-36}}{2\times 3}$$

Add 100 to -36.

$$v=\frac{-10±\sqrt{64}}{2\times 3}$$

Take the square root of 64.

$$v=\frac{-10±8}{2\times 3}$$

Multiply 2 times 3.

$$v=\frac{-10±8}{6}$$

Now solve the equation v=\frac{-10±8}{6} when ± is plus. Add -10 to 8.

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

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

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

Now solve the equation v=\frac{-10±8}{6} when ± is minus. Subtract 8 from -10.

$$v=-\frac{18}{6}$$

Divide -18 by 6.

$$v=-3$$

The equation is now solved.

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

Steps for Completing the Square

Add 10v to both sides.

$$3v^{2}+3+10v=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$3v^{2}+10v=-3$$

Divide both sides by 3.

$$\frac{3v^{2}+10v}{3}=-\frac{3}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$v^{2}+\frac{10}{3}v=-\frac{3}{3}$$

Divide -3 by 3.

$$v^{2}+\frac{10}{3}v=-1$$

Divide \frac{10}{3}, the coefficient of the x term, by 2 to get \frac{5}{3}. Then add the square of \frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+\frac{10}{3}v+\left(\frac{5}{3}\right)^{2}=-1+\left(\frac{5}{3}\right)^{2}$$

Square \frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$v^{2}+\frac{10}{3}v+\frac{25}{9}=-1+\frac{25}{9}$$

Add -1 to \frac{25}{9}.

$$v^{2}+\frac{10}{3}v+\frac{25}{9}=\frac{16}{9}$$

Factor v^{2}+\frac{10}{3}v+\frac{25}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+\frac{5}{3}\right)^{2}=\frac{16}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+\frac{5}{3}\right)^{2}}=\sqrt{\frac{16}{9}}$$

Simplify.

$$v+\frac{5}{3}=\frac{4}{3} v+\frac{5}{3}=-\frac{4}{3}$$

Subtract \frac{5}{3} from both sides of the equation.

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