Solve v/12=-4/v+14

$$\frac { v } { 12 } = \frac { - 4 } { v + 14 }$$

Solve for v

$$v=-8$$

$$v=-6$$

Steps Using the Quadratic Formula

Variable v cannot be equal to -14 since division by zero is not defined. Multiply both sides of the equation by 12\left(v+14\right), the least common multiple of 12,v+14.

$$\left(v+14\right)v=12\left(-4\right)$$

Use the distributive property to multiply v+14 by v.

$$v^{2}+14v=12\left(-4\right)$$

Multiply 12 and -4 to get -48.

$$v^{2}+14v=-48$$

Add 48 to both sides.

$$v^{2}+14v+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-14±\sqrt{14^{2}-4\times 48}}{2}$$

Square 14.

$$v=\frac{-14±\sqrt{196-4\times 48}}{2}$$

Multiply -4 times 48.

$$v=\frac{-14±\sqrt{196-192}}{2}$$

Add 196 to -192.

$$v=\frac{-14±\sqrt{4}}{2}$$

Take the square root of 4.

$$v=\frac{-14±2}{2}$$

Now solve the equation v=\frac{-14±2}{2} when ± is plus. Add -14 to 2.

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

Divide -12 by 2.

$$v=-6$$

Now solve the equation v=\frac{-14±2}{2} when ± is minus. Subtract 2 from -14.

$$v=-\frac{16}{2}$$

Divide -16 by 2.

$$v=-8$$

The equation is now solved.

$$v=-6 v=-8$$

Steps for Completing the Square

Variable v cannot be equal to -14 since division by zero is not defined. Multiply both sides of the equation by 12\left(v+14\right), the least common multiple of 12,v+14.

$$\left(v+14\right)v=12\left(-4\right)$$

Use the distributive property to multiply v+14 by v.

$$v^{2}+14v=12\left(-4\right)$$

Multiply 12 and -4 to get -48.

$$v^{2}+14v=-48$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+14v+7^{2}=-48+7^{2}$$

Square 7.

$$v^{2}+14v+49=-48+49$$

Add -48 to 49.

$$v^{2}+14v+49=1$$

Factor v^{2}+14v+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+7\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+7\right)^{2}}=\sqrt{1}$$

Simplify.

$$v+7=1 v+7=-1$$

Subtract 7 from both sides of the equation.

$$v=-6 v=-8$$