Solve 72+x=12/22+x

$$72 + x = \frac { 12 } { 22 + x }$$

Solve for x

$$x=7\sqrt{13}-47\approx -21.761141072$$

$$x=-7\sqrt{13}-47\approx -72.238858928$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -22 since division by zero is not defined. Multiply both sides of the equation by x+22.

$$\left(x+22\right)\times 72+\left(x+22\right)x=12$$

Use the distributive property to multiply x+22 by 72.

$$72x+1584+\left(x+22\right)x=12$$

Use the distributive property to multiply x+22 by x.

$$72x+1584+x^{2}+22x=12$$

Combine 72x and 22x to get 94x.

$$94x+1584+x^{2}=12$$

Subtract 12 from both sides.

$$94x+1584+x^{2}-12=0$$

Subtract 12 from 1584 to get 1572.

$$94x+1572+x^{2}=0$$

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.

$$x^{2}+94x+1572=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 94 for b, and 1572 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-94±\sqrt{94^{2}-4\times 1572}}{2}$$

Square 94.

$$x=\frac{-94±\sqrt{8836-4\times 1572}}{2}$$

Multiply -4 times 1572.

$$x=\frac{-94±\sqrt{8836-6288}}{2}$$

Add 8836 to -6288.

$$x=\frac{-94±\sqrt{2548}}{2}$$

Take the square root of 2548.

$$x=\frac{-94±14\sqrt{13}}{2}$$

Now solve the equation x=\frac{-94±14\sqrt{13}}{2} when ± is plus. Add -94 to 14\sqrt{13}.

$$x=\frac{14\sqrt{13}-94}{2}$$

Divide -94+14\sqrt{13} by 2.

$$x=7\sqrt{13}-47$$

Now solve the equation x=\frac{-94±14\sqrt{13}}{2} when ± is minus. Subtract 14\sqrt{13} from -94.

$$x=\frac{-14\sqrt{13}-94}{2}$$

Divide -94-14\sqrt{13} by 2.

$$x=-7\sqrt{13}-47$$

The equation is now solved.

$$x=7\sqrt{13}-47 x=-7\sqrt{13}-47$$

Steps for Completing the Square

Variable x cannot be equal to -22 since division by zero is not defined. Multiply both sides of the equation by x+22.

$$\left(x+22\right)\times 72+\left(x+22\right)x=12$$

Use the distributive property to multiply x+22 by 72.

$$72x+1584+\left(x+22\right)x=12$$

Use the distributive property to multiply x+22 by x.

$$72x+1584+x^{2}+22x=12$$

Combine 72x and 22x to get 94x.

$$94x+1584+x^{2}=12$$

Subtract 1584 from both sides.

$$94x+x^{2}=12-1584$$

Subtract 1584 from 12 to get -1572.

$$94x+x^{2}=-1572$$

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.

$$x^{2}+94x=-1572$$

Divide 94, the coefficient of the x term, by 2 to get 47. Then add the square of 47 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+94x+47^{2}=-1572+47^{2}$$

Square 47.

$$x^{2}+94x+2209=-1572+2209$$

Add -1572 to 2209.

$$x^{2}+94x+2209=637$$

Factor x^{2}+94x+2209. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+47\right)^{2}=637$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+47\right)^{2}}=\sqrt{637}$$

Simplify.

$$x+47=7\sqrt{13} x+47=-7\sqrt{13}$$

Subtract 47 from both sides of the equation.

$$x=7\sqrt{13}-47 x=-7\sqrt{13}-47$$