Solve 5/x=1+3/x+2

$$\frac { 5 } { x } = 1 + \frac { 3 } { x + 2 }$$

Solve for x

$$x=\sqrt{10}\approx 3.16227766$$

$$x=-\sqrt{10}\approx -3.16227766$$

Steps by Finding Square Root

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,x+2.

$$\left(x+2\right)\times 5=x\left(x+2\right)+x\times 3$$

Use the distributive property to multiply x+2 by 5.

$$5x+10=x\left(x+2\right)+x\times 3$$

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

$$5x+10=x^{2}+2x+x\times 3$$

Combine 2x and x\times 3 to get 5x.

$$5x+10=x^{2}+5x$$

Subtract x^{2} from both sides.

$$5x+10-x^{2}=5x$$

Subtract 5x from both sides.

$$5x+10-x^{2}-5x=0$$

Combine 5x and -5x to get 0.

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

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$-x^{2}=-10$$

Divide both sides by -1.

$$x^{2}=\frac{-10}{-1}$$

Fraction \frac{-10}{-1} can be simplified to 10 by removing the negative sign from both the numerator and the denominator.

$$x^{2}=10$$

Take the square root of both sides of the equation.

$$x=\sqrt{10} x=-\sqrt{10}$$

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,x+2.

$$\left(x+2\right)\times 5=x\left(x+2\right)+x\times 3$$

Use the distributive property to multiply x+2 by 5.

$$5x+10=x\left(x+2\right)+x\times 3$$

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

$$5x+10=x^{2}+2x+x\times 3$$

Combine 2x and x\times 3 to get 5x.

$$5x+10=x^{2}+5x$$

Subtract x^{2} from both sides.

$$5x+10-x^{2}=5x$$

Subtract 5x from both sides.

$$5x+10-x^{2}-5x=0$$

Combine 5x and -5x to get 0.

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

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4\times 10}}{2\left(-1\right)}$$

Multiply 4 times 10.

$$x=\frac{0±\sqrt{40}}{2\left(-1\right)}$$

Take the square root of 40.

$$x=\frac{0±2\sqrt{10}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±2\sqrt{10}}{-2}$$

Now solve the equation x=\frac{0±2\sqrt{10}}{-2} when ± is plus.

$$x=-\sqrt{10}$$

Now solve the equation x=\frac{0±2\sqrt{10}}{-2} when ± is minus.

$$x=\sqrt{10}$$

The equation is now solved.

$$x=-\sqrt{10} x=\sqrt{10}$$