Solve 22+x/30=60/22+2x

$$\frac { 22 + x } { 30 } = \frac { 60 } { 22 + 2 x }$$

Solve for x

$$x=-47$$

$$x=14$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -11 since division by zero is not defined. Multiply both sides of the equation by 30\left(x+11\right), the least common multiple of 30,22+2x.

$$\left(x+11\right)\left(22+x\right)=15\times 60$$

Use the distributive property to multiply x+11 by 22+x and combine like terms.

$$33x+x^{2}+242=15\times 60$$

Multiply 15 and 60 to get 900.

$$33x+x^{2}+242=900$$

Subtract 900 from both sides.

$$33x+x^{2}+242-900=0$$

Subtract 900 from 242 to get -658.

$$33x+x^{2}-658=0$$

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.

$$x^{2}+33x-658=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 33 for b, and -658 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-33±\sqrt{33^{2}-4\left(-658\right)}}{2}$$

Square 33.

$$x=\frac{-33±\sqrt{1089-4\left(-658\right)}}{2}$$

Multiply -4 times -658.

$$x=\frac{-33±\sqrt{1089+2632}}{2}$$

Add 1089 to 2632.

$$x=\frac{-33±\sqrt{3721}}{2}$$

Take the square root of 3721.

$$x=\frac{-33±61}{2}$$

Now solve the equation x=\frac{-33±61}{2} when ± is plus. Add -33 to 61.

$$x=\frac{28}{2}$$

Divide 28 by 2.

$$x=14$$

Now solve the equation x=\frac{-33±61}{2} when ± is minus. Subtract 61 from -33.

$$x=-\frac{94}{2}$$

Divide -94 by 2.

$$x=-47$$

The equation is now solved.

$$x=14 x=-47$$

Steps for Completing the Square

Variable x cannot be equal to -11 since division by zero is not defined. Multiply both sides of the equation by 30\left(x+11\right), the least common multiple of 30,22+2x.

$$\left(x+11\right)\left(22+x\right)=15\times 60$$

Use the distributive property to multiply x+11 by 22+x and combine like terms.

$$33x+x^{2}+242=15\times 60$$

Multiply 15 and 60 to get 900.

$$33x+x^{2}+242=900$$

Subtract 242 from both sides.

$$33x+x^{2}=900-242$$

Subtract 242 from 900 to get 658.

$$33x+x^{2}=658$$

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.

$$x^{2}+33x=658$$

Divide 33, the coefficient of the x term, by 2 to get \frac{33}{2}. Then add the square of \frac{33}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+33x+\left(\frac{33}{2}\right)^{2}=658+\left(\frac{33}{2}\right)^{2}$$

Square \frac{33}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+33x+\frac{1089}{4}=658+\frac{1089}{4}$$

Add 658 to \frac{1089}{4}.

$$x^{2}+33x+\frac{1089}{4}=\frac{3721}{4}$$

Factor x^{2}+33x+\frac{1089}{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}.

$$\left(x+\frac{33}{2}\right)^{2}=\frac{3721}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{33}{2}\right)^{2}}=\sqrt{\frac{3721}{4}}$$

Simplify.

$$x+\frac{33}{2}=\frac{61}{2} x+\frac{33}{2}=-\frac{61}{2}$$

Subtract \frac{33}{2} from both sides of the equation.

$$x=14 x=-47$$