Solve 22-x/24=2/24-x

$$\frac{ 22-x }{ 24 } = \frac{ 2 }{ 24-x }$$

Solve for x

$$x=16$$

$$x=30$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 24 since division by zero is not defined. Multiply both sides of the equation by 24\left(x-24\right), the least common multiple of 24,24-x.

$$\left(x-24\right)\left(22-x\right)=-24\times 2$$

Use the distributive property to multiply x-24 by 22-x and combine like terms.

$$46x-x^{2}-528=-24\times 2$$

Multiply -24 and 2 to get -48.

$$46x-x^{2}-528=-48$$

Add 48 to both sides.

$$46x-x^{2}-528+48=0$$

Add -528 and 48 to get -480.

$$46x-x^{2}-480=0$$

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.

$$-x^{2}+46x-480=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 46 for b, and -480 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-46±\sqrt{46^{2}-4\left(-1\right)\left(-480\right)}}{2\left(-1\right)}$$

Square 46.

$$x=\frac{-46±\sqrt{2116-4\left(-1\right)\left(-480\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-46±\sqrt{2116+4\left(-480\right)}}{2\left(-1\right)}$$

Multiply 4 times -480.

$$x=\frac{-46±\sqrt{2116-1920}}{2\left(-1\right)}$$

Add 2116 to -1920.

$$x=\frac{-46±\sqrt{196}}{2\left(-1\right)}$$

Take the square root of 196.

$$x=\frac{-46±14}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-46±14}{-2}$$

Now solve the equation x=\frac{-46±14}{-2} when ± is plus. Add -46 to 14.

$$x=-\frac{32}{-2}$$

Divide -32 by -2.

$$x=16$$

Now solve the equation x=\frac{-46±14}{-2} when ± is minus. Subtract 14 from -46.

$$x=-\frac{60}{-2}$$

Divide -60 by -2.

$$x=30$$

The equation is now solved.

$$x=16 x=30$$

Steps for Completing the Square

Variable x cannot be equal to 24 since division by zero is not defined. Multiply both sides of the equation by 24\left(x-24\right), the least common multiple of 24,24-x.

$$\left(x-24\right)\left(22-x\right)=-24\times 2$$

Use the distributive property to multiply x-24 by 22-x and combine like terms.

$$46x-x^{2}-528=-24\times 2$$

Multiply -24 and 2 to get -48.

$$46x-x^{2}-528=-48$$

Add 528 to both sides.

$$46x-x^{2}=-48+528$$

Add -48 and 528 to get 480.

$$46x-x^{2}=480$$

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.

$$-x^{2}+46x=480$$

Divide both sides by -1.

$$\frac{-x^{2}+46x}{-1}=\frac{480}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{46}{-1}x=\frac{480}{-1}$$

Divide 46 by -1.

$$x^{2}-46x=\frac{480}{-1}$$

Divide 480 by -1.

$$x^{2}-46x=-480$$

Divide -46, the coefficient of the x term, by 2 to get -23. Then add the square of -23 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-46x+\left(-23\right)^{2}=-480+\left(-23\right)^{2}$$

Square -23.

$$x^{2}-46x+529=-480+529$$

Add -480 to 529.

$$x^{2}-46x+529=49$$

Factor x^{2}-46x+529. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-23\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-23\right)^{2}}=\sqrt{49}$$

Simplify.

$$x-23=7 x-23=-7$$

Add 23 to both sides of the equation.

$$x=30 x=16$$