Solve x+2=28/x+1

$$x + 2 = \frac { 28 } { x + 1 }$$

Solve for x

$$x = \frac{\sqrt{113} - 3}{2} \approx 3.815072906$$

$$x=\frac{-\sqrt{113}-3}{2}\approx -6.815072906$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$\left(x+1\right)x+\left(x+1\right)\times 2=28$$

Use the distributive property to multiply x+1 by x.

$$x^{2}+x+\left(x+1\right)\times 2=28$$

Use the distributive property to multiply x+1 by 2.

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

Combine x and 2x to get 3x.

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

Subtract 28 from both sides.

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

Subtract 28 from 2 to get -26.

$$x^{2}+3x-26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and -26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-26\right)}}{2}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-26\right)}}{2}$$

Multiply -4 times -26.

$$x=\frac{-3±\sqrt{9+104}}{2}$$

Add 9 to 104.

$$x=\frac{-3±\sqrt{113}}{2}$$

Now solve the equation x=\frac{-3±\sqrt{113}}{2} when ± is plus. Add -3 to \sqrt{113}.

$$x=\frac{\sqrt{113}-3}{2}$$

Now solve the equation x=\frac{-3±\sqrt{113}}{2} when ± is minus. Subtract \sqrt{113} from -3.

$$x=\frac{-\sqrt{113}-3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{113}-3}{2} x=\frac{-\sqrt{113}-3}{2}$$

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$\left(x+1\right)x+\left(x+1\right)\times 2=28$$

Use the distributive property to multiply x+1 by x.

$$x^{2}+x+\left(x+1\right)\times 2=28$$

Use the distributive property to multiply x+1 by 2.

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

Combine x and 2x to get 3x.

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

Subtract 2 from both sides.

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

Subtract 2 from 28 to get 26.

$$x^{2}+3x=26$$

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.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=26+\left(\frac{3}{2}\right)^{2}$$

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

$$x^{2}+3x+\frac{9}{4}=26+\frac{9}{4}$$

Add 26 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{113}{4}$$

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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{113}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{113}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{113}}{2} x+\frac{3}{2}=-\frac{\sqrt{113}}{2}$$

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

$$x=\frac{\sqrt{113}-3}{2} x=\frac{-\sqrt{113}-3}{2}$$