Solve 3v^2-6=-7v

$$3 v ^ { 2 } - 6 = - 7 v$$

Solve for v

$$v=-3$$

$$v=\frac{2}{3}\approx 0.666666667$$

Steps Using Factoring By Grouping

Add 7v to both sides.

$$3v^{2}-6+7v=0$$

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

$$3v^{2}+7v-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3v^{2}+av+bv-6. To find a and b, set up a system to be solved.

$$a+b=7 ab=3\left(-6\right)=-18$$

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 -18.

$$-1,18 -2,9 -3,6$$

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 7.

$$a=-2 b=9$$

Rewrite 3v^{2}+7v-6 as \left(3v^{2}-2v\right)+\left(9v-6\right).

$$\left(3v^{2}-2v\right)+\left(9v-6\right)$$

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

$$v\left(3v-2\right)+3\left(3v-2\right)$$

Factor out common term 3v-2 by using distributive property.

$$\left(3v-2\right)\left(v+3\right)$$

To find equation solutions, solve 3v-2=0 and v+3=0.

$$v=\frac{2}{3} v=-3$$

Steps Using the Quadratic Formula

Add 7v to both sides.

$$3v^{2}-6+7v=0$$

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.

$$3v^{2}+7v-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 7 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-7±\sqrt{7^{2}-4\times 3\left(-6\right)}}{2\times 3}$$

Square 7.

$$v=\frac{-7±\sqrt{49-4\times 3\left(-6\right)}}{2\times 3}$$

Multiply -4 times 3.

$$v=\frac{-7±\sqrt{49-12\left(-6\right)}}{2\times 3}$$

Multiply -12 times -6.

$$v=\frac{-7±\sqrt{49+72}}{2\times 3}$$

Add 49 to 72.

$$v=\frac{-7±\sqrt{121}}{2\times 3}$$

Take the square root of 121.

$$v=\frac{-7±11}{2\times 3}$$

Multiply 2 times 3.

$$v=\frac{-7±11}{6}$$

Now solve the equation v=\frac{-7±11}{6} when ± is plus. Add -7 to 11.

$$v=\frac{4}{6}$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$v=\frac{2}{3}$$

Now solve the equation v=\frac{-7±11}{6} when ± is minus. Subtract 11 from -7.

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

Divide -18 by 6.

$$v=-3$$

The equation is now solved.

$$v=\frac{2}{3} v=-3$$

Steps for Completing the Square

Add 7v to both sides.

$$3v^{2}-6+7v=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$3v^{2}+7v=6$$

Divide both sides by 3.

$$\frac{3v^{2}+7v}{3}=\frac{6}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$v^{2}+\frac{7}{3}v=\frac{6}{3}$$

Divide 6 by 3.

$$v^{2}+\frac{7}{3}v=2$$

Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+\frac{7}{3}v+\left(\frac{7}{6}\right)^{2}=2+\left(\frac{7}{6}\right)^{2}$$

Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$v^{2}+\frac{7}{3}v+\frac{49}{36}=2+\frac{49}{36}$$

Add 2 to \frac{49}{36}.

$$v^{2}+\frac{7}{3}v+\frac{49}{36}=\frac{121}{36}$$

Factor v^{2}+\frac{7}{3}v+\frac{49}{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}.

$$\left(v+\frac{7}{6}\right)^{2}=\frac{121}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+\frac{7}{6}\right)^{2}}=\sqrt{\frac{121}{36}}$$

Simplify.

$$v+\frac{7}{6}=\frac{11}{6} v+\frac{7}{6}=-\frac{11}{6}$$

Subtract \frac{7}{6} from both sides of the equation.

$$v=\frac{2}{3} v=-3$$