Solve 24/x+2+1=24/x

$$\frac{ 24 }{ x+2 } +1= \frac{ 24 }{ x }$$

Solve for x

$$x=-8$$

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x+2,x.

$$x\times 24+x\left(x+2\right)=\left(x+2\right)\times 24$$

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

$$x\times 24+x^{2}+2x=\left(x+2\right)\times 24$$

Combine x\times 24 and 2x to get 26x.

$$26x+x^{2}=\left(x+2\right)\times 24$$

Use the distributive property to multiply x+2 by 24.

$$26x+x^{2}=24x+48$$

Subtract 24x from both sides.

$$26x+x^{2}-24x=48$$

Combine 26x and -24x to get 2x.

$$2x+x^{2}=48$$

Subtract 48 from both sides.

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

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.

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

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

$$x=\frac{-2±\sqrt{2^{2}-4\left(-48\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-48\right)}}{2}$$

Multiply -4 times -48.

$$x=\frac{-2±\sqrt{4+192}}{2}$$

Add 4 to 192.

$$x=\frac{-2±\sqrt{196}}{2}$$

Take the square root of 196.

$$x=\frac{-2±14}{2}$$

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

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

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

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

Divide -16 by 2.

$$x=-8$$

The equation is now solved.

$$x=6 x=-8$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x+2,x.

$$x\times 24+x\left(x+2\right)=\left(x+2\right)\times 24$$

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

$$x\times 24+x^{2}+2x=\left(x+2\right)\times 24$$

Combine x\times 24 and 2x to get 26x.

$$26x+x^{2}=\left(x+2\right)\times 24$$

Use the distributive property to multiply x+2 by 24.

$$26x+x^{2}=24x+48$$

Subtract 24x from both sides.

$$26x+x^{2}-24x=48$$

Combine 26x and -24x to get 2x.

$$2x+x^{2}=48$$

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.

$$x^{2}+2x=48$$

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.

$$x^{2}+2x+1^{2}=48+1^{2}$$

Square 1.

$$x^{2}+2x+1=48+1$$

Add 48 to 1.

$$x^{2}+2x+1=49$$

Factor x^{2}+2x+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}.

$$\left(x+1\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{49}$$

Simplify.

$$x+1=7 x+1=-7$$

Subtract 1 from both sides of the equation.

$$x=6 x=-8$$