Solve -v+1/2=-6/v

$$\frac { - v + 1 } { 2 } = \frac { - 6 } { v }$$

Solve for v

$$v=-3$$

$$v=4$$

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 2v, the least common multiple of 2,v.

$$v\left(-v+1\right)=2\left(-6\right)$$

Use the distributive property to multiply v by -v+1.

$$v\left(-v\right)+v=2\left(-6\right)$$

Multiply 2 and -6 to get -12.

$$v\left(-v\right)+v=-12$$

Add 12 to both sides.

$$v\left(-v\right)+v+12=0$$

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

$$v^{2}\left(-1\right)+v+12=0$$

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.

$$-v^{2}+v+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Square 1.

$$v=\frac{-1±\sqrt{1-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$v=\frac{-1±\sqrt{1+4\times 12}}{2\left(-1\right)}$$

Multiply 4 times 12.

$$v=\frac{-1±\sqrt{1+48}}{2\left(-1\right)}$$

Add 1 to 48.

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

Take the square root of 49.

$$v=\frac{-1±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$v=\frac{-1±7}{-2}$$

Now solve the equation v=\frac{-1±7}{-2} when ± is plus. Add -1 to 7.

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

Divide 6 by -2.

$$v=-3$$

Now solve the equation v=\frac{-1±7}{-2} when ± is minus. Subtract 7 from -1.

$$v=-\frac{8}{-2}$$

Divide -8 by -2.

$$v=4$$

The equation is now solved.

$$v=-3 v=4$$

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 2v, the least common multiple of 2,v.

$$v\left(-v+1\right)=2\left(-6\right)$$

Use the distributive property to multiply v by -v+1.

$$v\left(-v\right)+v=2\left(-6\right)$$

Multiply 2 and -6 to get -12.

$$v\left(-v\right)+v=-12$$

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

$$v^{2}\left(-1\right)+v=-12$$

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.

$$-v^{2}+v=-12$$

Divide both sides by -1.

$$\frac{-v^{2}+v}{-1}=-\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$v^{2}+\frac{1}{-1}v=-\frac{12}{-1}$$

Divide 1 by -1.

$$v^{2}-v=-\frac{12}{-1}$$

Divide -12 by -1.

$$v^{2}-v=12$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-v+\left(-\frac{1}{2}\right)^{2}=12+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$v^{2}-v+\frac{1}{4}=12+\frac{1}{4}$$

Add 12 to \frac{1}{4}.

$$v^{2}-v+\frac{1}{4}=\frac{49}{4}$$

Factor v^{2}-v+\frac{1}{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}.

$$\left(v-\frac{1}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-\frac{1}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$v-\frac{1}{2}=\frac{7}{2} v-\frac{1}{2}=-\frac{7}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$v=4 v=-3$$