Solve 3=v-10/v

$$3 = v - \frac { 10 } { v }$$

Solve for v

$$v=-2$$

$$v=5$$

Steps Using the Quadratic Formula

Variable v cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by v.

$$3v=vv-10$$

Multiply v and v to get v^{2}.

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

Subtract v^{2} from both sides.

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

Add 10 to both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Square 3.

$$v=\frac{-3±\sqrt{9-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$v=\frac{-3±\sqrt{9+4\times 10}}{2\left(-1\right)}$$

Multiply 4 times 10.

$$v=\frac{-3±\sqrt{9+40}}{2\left(-1\right)}$$

Add 9 to 40.

$$v=\frac{-3±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of 49.

$$v=\frac{-3±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$v=\frac{-3±7}{-2}$$

Now solve the equation v=\frac{-3±7}{-2} when ± is plus. Add -3 to 7.

$$v=\frac{4}{-2}$$

Divide 4 by -2.

$$v=-2$$

Now solve the equation v=\frac{-3±7}{-2} when ± is minus. Subtract 7 from -3.

$$v=-\frac{10}{-2}$$

Divide -10 by -2.

$$v=5$$

The equation is now solved.

$$v=-2 v=5$$

Steps for Completing the Square

Variable v cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by v.

$$3v=vv-10$$

Multiply v and v to get v^{2}.

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

Subtract v^{2} from both sides.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 3 by -1.

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

Divide -10 by -1.

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

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

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$v^{2}-3v+\frac{9}{4}=10+\frac{9}{4}$$

Add 10 to \frac{9}{4}.

$$v^{2}-3v+\frac{9}{4}=\frac{49}{4}$$

Factor v^{2}-3v+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v-\frac{3}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$v-\frac{3}{2}=\frac{7}{2} v-\frac{3}{2}=-\frac{7}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$v=5 v=-2$$