Solve 10/xleft(x+2right)=10

$$\frac{ 10 }{ x \left( x+2 \right) } = 10$$

Solve for x (complex solution)

$$x=\sqrt{2}-1\approx 0.414213562$$

$$x=-\left(\sqrt{2}+1\right)\approx -2.414213562$$

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).

$$10=10x\left(x+2\right)$$

Use the distributive property to multiply 10x by x+2.

$$10=10x^{2}+20x$$

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

$$10x^{2}+20x=10$$

Subtract 10 from both sides.

$$10x^{2}+20x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 20 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\times 10\left(-10\right)}}{2\times 10}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\times 10\left(-10\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-20±\sqrt{400-40\left(-10\right)}}{2\times 10}$$

Multiply -40 times -10.

$$x=\frac{-20±\sqrt{400+400}}{2\times 10}$$

Add 400 to 400.

$$x=\frac{-20±\sqrt{800}}{2\times 10}$$

Take the square root of 800.

$$x=\frac{-20±20\sqrt{2}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-20±20\sqrt{2}}{20}$$

Now solve the equation x=\frac{-20±20\sqrt{2}}{20} when ± is plus. Add -20 to 20\sqrt{2}.

$$x=\frac{20\sqrt{2}-20}{20}$$

Divide -20+20\sqrt{2} by 20.

$$x=\sqrt{2}-1$$

Now solve the equation x=\frac{-20±20\sqrt{2}}{20} when ± is minus. Subtract 20\sqrt{2} from -20.

$$x=\frac{-20\sqrt{2}-20}{20}$$

Divide -20-20\sqrt{2} by 20.

$$x=-\sqrt{2}-1$$

The equation is now solved.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$

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).

$$10=10x\left(x+2\right)$$

Use the distributive property to multiply 10x by x+2.

$$10=10x^{2}+20x$$

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

$$10x^{2}+20x=10$$

Divide both sides by 10.

$$\frac{10x^{2}+20x}{10}=\frac{10}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{20}{10}x=\frac{10}{10}$$

Divide 20 by 10.

$$x^{2}+2x=\frac{10}{10}$$

Divide 10 by 10.

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

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.

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

Square 1.

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

Add 1 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=\sqrt{2} x+1=-\sqrt{2}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$

Solve for x

$$x=\sqrt{2}-1\approx 0.414213562$$

$$x=-\sqrt{2}-1\approx -2.414213562$$

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).

$$10=10x\left(x+2\right)$$

Use the distributive property to multiply 10x by x+2.

$$10=10x^{2}+20x$$

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

$$10x^{2}+20x=10$$

Subtract 10 from both sides.

$$10x^{2}+20x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 20 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\times 10\left(-10\right)}}{2\times 10}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\times 10\left(-10\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-20±\sqrt{400-40\left(-10\right)}}{2\times 10}$$

Multiply -40 times -10.

$$x=\frac{-20±\sqrt{400+400}}{2\times 10}$$

Add 400 to 400.

$$x=\frac{-20±\sqrt{800}}{2\times 10}$$

Take the square root of 800.

$$x=\frac{-20±20\sqrt{2}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-20±20\sqrt{2}}{20}$$

Now solve the equation x=\frac{-20±20\sqrt{2}}{20} when ± is plus. Add -20 to 20\sqrt{2}.

$$x=\frac{20\sqrt{2}-20}{20}$$

Divide -20+20\sqrt{2} by 20.

$$x=\sqrt{2}-1$$

Now solve the equation x=\frac{-20±20\sqrt{2}}{20} when ± is minus. Subtract 20\sqrt{2} from -20.

$$x=\frac{-20\sqrt{2}-20}{20}$$

Divide -20-20\sqrt{2} by 20.

$$x=-\sqrt{2}-1$$

The equation is now solved.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$

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).

$$10=10x\left(x+2\right)$$

Use the distributive property to multiply 10x by x+2.

$$10=10x^{2}+20x$$

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

$$10x^{2}+20x=10$$

Divide both sides by 10.

$$\frac{10x^{2}+20x}{10}=\frac{10}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{20}{10}x=\frac{10}{10}$$

Divide 20 by 10.

$$x^{2}+2x=\frac{10}{10}$$

Divide 10 by 10.

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

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.

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

Square 1.

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

Add 1 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=\sqrt{2} x+1=-\sqrt{2}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$