Solve 3v^2-15=6v

$$3 v ^ { 2 } - 15 = 6 v$$

Solve for v

$$v=\sqrt{6}+1\approx 3.449489743$$

$$v=1-\sqrt{6}\approx -1.449489743$$

Steps Using the Quadratic Formula

Subtract 6v from both sides.

$$3v^{2}-15-6v=0$$

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.

$$3v^{2}-6v-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -6 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3\left(-15\right)}}{2\times 3}$$

Square -6.

$$v=\frac{-\left(-6\right)±\sqrt{36-4\times 3\left(-15\right)}}{2\times 3}$$

Multiply -4 times 3.

$$v=\frac{-\left(-6\right)±\sqrt{36-12\left(-15\right)}}{2\times 3}$$

Multiply -12 times -15.

$$v=\frac{-\left(-6\right)±\sqrt{36+180}}{2\times 3}$$

Add 36 to 180.

$$v=\frac{-\left(-6\right)±\sqrt{216}}{2\times 3}$$

Take the square root of 216.

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

The opposite of -6 is 6.

$$v=\frac{6±6\sqrt{6}}{2\times 3}$$

Multiply 2 times 3.

$$v=\frac{6±6\sqrt{6}}{6}$$

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

$$v=\frac{6\sqrt{6}+6}{6}$$

Divide 6+6\sqrt{6} by 6.

$$v=\sqrt{6}+1$$

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

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

Divide 6-6\sqrt{6} by 6.

$$v=1-\sqrt{6}$$

The equation is now solved.

$$v=\sqrt{6}+1 v=1-\sqrt{6}$$

Steps for Completing the Square

Subtract 6v from both sides.

$$3v^{2}-15-6v=0$$

Add 15 to both sides. Anything plus zero gives itself.

$$3v^{2}-6v=15$$

Divide both sides by 3.

$$\frac{3v^{2}-6v}{3}=\frac{15}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$v^{2}+\left(-\frac{6}{3}\right)v=\frac{15}{3}$$

Divide -6 by 3.

$$v^{2}-2v=\frac{15}{3}$$

Divide 15 by 3.

$$v^{2}-2v=5$$

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.

$$v^{2}-2v+1=5+1$$

Add 5 to 1.

$$v^{2}-2v+1=6$$

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}.

$$\left(v-1\right)^{2}=6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-1\right)^{2}}=\sqrt{6}$$

Simplify.

$$v-1=\sqrt{6} v-1=-\sqrt{6}$$

Add 1 to both sides of the equation.

$$v=\sqrt{6}+1 v=1-\sqrt{6}$$