Solve 12/r+2=r/2

$$\frac { 12 } { r + 2 } = \frac { r } { 2 }$$

Solve for r

$$r=-6$$

$$r=4$$

Steps Using the Quadratic Formula

Variable r cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by 2\left(r+2\right), the least common multiple of r+2,2.

$$2\times 12=\left(r+2\right)r$$

Multiply 2 and 12 to get 24.

$$24=\left(r+2\right)r$$

Use the distributive property to multiply r+2 by r.

$$24=r^{2}+2r$$

Swap sides so that all variable terms are on the left hand side.

$$r^{2}+2r=24$$

Subtract 24 from both sides.

$$r^{2}+2r-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-2±\sqrt{2^{2}-4\left(-24\right)}}{2}$$

Square 2.

$$r=\frac{-2±\sqrt{4-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$r=\frac{-2±\sqrt{4+96}}{2}$$

Add 4 to 96.

$$r=\frac{-2±\sqrt{100}}{2}$$

Take the square root of 100.

$$r=\frac{-2±10}{2}$$

Now solve the equation r=\frac{-2±10}{2} when ± is plus. Add -2 to 10.

$$r=\frac{8}{2}$$

Divide 8 by 2.

$$r=4$$

Now solve the equation r=\frac{-2±10}{2} when ± is minus. Subtract 10 from -2.

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

Divide -12 by 2.

$$r=-6$$

The equation is now solved.

$$r=4 r=-6$$

Steps for Completing the Square

Variable r cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by 2\left(r+2\right), the least common multiple of r+2,2.

$$2\times 12=\left(r+2\right)r$$

Multiply 2 and 12 to get 24.

$$24=\left(r+2\right)r$$

Use the distributive property to multiply r+2 by r.

$$24=r^{2}+2r$$

Swap sides so that all variable terms are on the left hand side.

$$r^{2}+2r=24$$

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.

$$r^{2}+2r+1^{2}=24+1^{2}$$

Square 1.

$$r^{2}+2r+1=24+1$$

Add 24 to 1.

$$r^{2}+2r+1=25$$

Factor r^{2}+2r+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}.

$$\left(r+1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+1\right)^{2}}=\sqrt{25}$$

Simplify.

$$r+1=5 r+1=-5$$

Subtract 1 from both sides of the equation.

$$r=4 r=-6$$