Solve x+1/3=10/x+2

$$\frac{ x+1 }{ 3 } = \frac{ 10 }{ x+2 }$$

Solve for x

$$x=-7$$

$$x=4$$

Steps Using the Quadratic Formula

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

$$\left(x+2\right)\left(x+1\right)=3\times 10$$

Use the distributive property to multiply x+2 by x+1 and combine like terms.

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

Multiply 3 and 10 to get 30.

$$x^{2}+3x+2=30$$

Subtract 30 from both sides.

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

Subtract 30 from 2 to get -28.

$$x^{2}+3x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-28\right)}}{2}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-28\right)}}{2}$$

Multiply -4 times -28.

$$x=\frac{-3±\sqrt{9+112}}{2}$$

Add 9 to 112.

$$x=\frac{-3±\sqrt{121}}{2}$$

Take the square root of 121.

$$x=\frac{-3±11}{2}$$

Now solve the equation x=\frac{-3±11}{2} when ± is plus. Add -3 to 11.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

Now solve the equation x=\frac{-3±11}{2} when ± is minus. Subtract 11 from -3.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

The equation is now solved.

$$x=4 x=-7$$

Steps for Completing the Square

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

$$\left(x+2\right)\left(x+1\right)=3\times 10$$

Use the distributive property to multiply x+2 by x+1 and combine like terms.

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

Multiply 3 and 10 to get 30.

$$x^{2}+3x+2=30$$

Subtract 2 from both sides.

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

Subtract 2 from 30 to get 28.

$$x^{2}+3x=28$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=28+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=28+\frac{9}{4}$$

Add 28 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{121}{4}$$

Factor x^{2}+3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{11}{2} x+\frac{3}{2}=-\frac{11}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=4 x=-7$$