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

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

Solve for x

$$x=\frac{\sqrt{10}-10}{3}\approx -2.27924078$$

$$x=\frac{-\sqrt{10}-10}{3}\approx -4.387425887$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,-2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right)\left(x+3\right), the least common multiple of x+2,x+3,x.

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

Use the distributive property to multiply x by x+3.

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

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

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Combine 3x and 2x to get 5x.

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

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$2x^{2}+5x=\left(x^{2}+5x+6\right)\times 5$$

Use the distributive property to multiply x^{2}+5x+6 by 5.

$$2x^{2}+5x=5x^{2}+25x+30$$

Subtract 5x^{2} from both sides.

$$2x^{2}+5x-5x^{2}=25x+30$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}+5x=25x+30$$

Subtract 25x from both sides.

$$-3x^{2}+5x-25x=30$$

Combine 5x and -25x to get -20x.

$$-3x^{2}-20x=30$$

Subtract 30 from both sides.

$$-3x^{2}-20x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -20 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\left(-3\right)\left(-30\right)}}{2\left(-3\right)}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\left(-3\right)\left(-30\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-20\right)±\sqrt{400+12\left(-30\right)}}{2\left(-3\right)}$$

Multiply 12 times -30.

$$x=\frac{-\left(-20\right)±\sqrt{400-360}}{2\left(-3\right)}$$

Add 400 to -360.

$$x=\frac{-\left(-20\right)±\sqrt{40}}{2\left(-3\right)}$$

Take the square root of 40.

$$x=\frac{-\left(-20\right)±2\sqrt{10}}{2\left(-3\right)}$$

The opposite of -20 is 20.

$$x=\frac{20±2\sqrt{10}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{20±2\sqrt{10}}{-6}$$

Now solve the equation x=\frac{20±2\sqrt{10}}{-6} when ± is plus. Add 20 to 2\sqrt{10}.

$$x=\frac{2\sqrt{10}+20}{-6}$$

Divide 20+2\sqrt{10} by -6.

$$x=\frac{-\sqrt{10}-10}{3}$$

Now solve the equation x=\frac{20±2\sqrt{10}}{-6} when ± is minus. Subtract 2\sqrt{10} from 20.

$$x=\frac{20-2\sqrt{10}}{-6}$$

Divide 20-2\sqrt{10} by -6.

$$x=\frac{\sqrt{10}-10}{3}$$

The equation is now solved.

$$x=\frac{-\sqrt{10}-10}{3} x=\frac{\sqrt{10}-10}{3}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,-2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right)\left(x+3\right), the least common multiple of x+2,x+3,x.

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

Use the distributive property to multiply x by x+3.

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

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

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Combine 3x and 2x to get 5x.

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

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$2x^{2}+5x=\left(x^{2}+5x+6\right)\times 5$$

Use the distributive property to multiply x^{2}+5x+6 by 5.

$$2x^{2}+5x=5x^{2}+25x+30$$

Subtract 5x^{2} from both sides.

$$2x^{2}+5x-5x^{2}=25x+30$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}+5x=25x+30$$

Subtract 25x from both sides.

$$-3x^{2}+5x-25x=30$$

Combine 5x and -25x to get -20x.

$$-3x^{2}-20x=30$$

Divide both sides by -3.

$$\frac{-3x^{2}-20x}{-3}=\frac{30}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{20}{-3}\right)x=\frac{30}{-3}$$

Divide -20 by -3.

$$x^{2}+\frac{20}{3}x=\frac{30}{-3}$$

Divide 30 by -3.

$$x^{2}+\frac{20}{3}x=-10$$

Divide \frac{20}{3}, the coefficient of the x term, by 2 to get \frac{10}{3}. Then add the square of \frac{10}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{20}{3}x+\left(\frac{10}{3}\right)^{2}=-10+\left(\frac{10}{3}\right)^{2}$$

Square \frac{10}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{20}{3}x+\frac{100}{9}=-10+\frac{100}{9}$$

Add -10 to \frac{100}{9}.

$$x^{2}+\frac{20}{3}x+\frac{100}{9}=\frac{10}{9}$$

Factor x^{2}+\frac{20}{3}x+\frac{100}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{10}{3}\right)^{2}=\frac{10}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{10}{3}\right)^{2}}=\sqrt{\frac{10}{9}}$$

Simplify.

$$x+\frac{10}{3}=\frac{\sqrt{10}}{3} x+\frac{10}{3}=-\frac{\sqrt{10}}{3}$$

Subtract \frac{10}{3} from both sides of the equation.

$$x=\frac{\sqrt{10}-10}{3} x=\frac{-\sqrt{10}-10}{3}$$