Solve 1/a+1-1/a+2=1/4

$$\frac { 1 } { a + 1 } - \frac { 1 } { a + 2 } = \frac { 1 } { 4 }$$

Solve for a

$$a=\frac{\sqrt{17}-3}{2}\approx 0.561552813$$

$$a=\frac{-\sqrt{17}-3}{2}\approx -3.561552813$$

Steps Using the Quadratic Formula

Variable a cannot be equal to any of the values -2,-1 since division by zero is not defined. Multiply both sides of the equation by 4\left(a+1\right)\left(a+2\right), the least common multiple of a+1,a+2,4.

$$4a+8-\left(4a+4\right)=\left(a+1\right)\left(a+2\right)$$

To find the opposite of 4a+4, find the opposite of each term.

$$4a+8-4a-4=\left(a+1\right)\left(a+2\right)$$

Combine 4a and -4a to get 0.

$$8-4=\left(a+1\right)\left(a+2\right)$$

Subtract 4 from 8 to get 4.

$$4=\left(a+1\right)\left(a+2\right)$$

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

$$4=a^{2}+3a+2$$

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

$$a^{2}+3a+2=4$$

Subtract 4 from both sides.

$$a^{2}+3a+2-4=0$$

Subtract 4 from 2 to get -2.

$$a^{2}+3a-2=0$$

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

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

Square 3.

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

Multiply -4 times -2.

$$a=\frac{-3±\sqrt{9+8}}{2}$$

Add 9 to 8.

$$a=\frac{-3±\sqrt{17}}{2}$$

Now solve the equation a=\frac{-3±\sqrt{17}}{2} when ± is plus. Add -3 to \sqrt{17}.

$$a=\frac{\sqrt{17}-3}{2}$$

Now solve the equation a=\frac{-3±\sqrt{17}}{2} when ± is minus. Subtract \sqrt{17} from -3.

$$a=\frac{-\sqrt{17}-3}{2}$$

The equation is now solved.

$$a=\frac{\sqrt{17}-3}{2} a=\frac{-\sqrt{17}-3}{2}$$

Steps for Completing the Square

Variable a cannot be equal to any of the values -2,-1 since division by zero is not defined. Multiply both sides of the equation by 4\left(a+1\right)\left(a+2\right), the least common multiple of a+1,a+2,4.

$$4a+8-\left(4a+4\right)=\left(a+1\right)\left(a+2\right)$$

To find the opposite of 4a+4, find the opposite of each term.

$$4a+8-4a-4=\left(a+1\right)\left(a+2\right)$$

Combine 4a and -4a to get 0.

$$8-4=\left(a+1\right)\left(a+2\right)$$

Subtract 4 from 8 to get 4.

$$4=\left(a+1\right)\left(a+2\right)$$

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

$$4=a^{2}+3a+2$$

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

$$a^{2}+3a+2=4$$

Subtract 2 from both sides.

$$a^{2}+3a=4-2$$

Subtract 2 from 4 to get 2.

$$a^{2}+3a=2$$

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.

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

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

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

Add 2 to \frac{9}{4}.

$$a^{2}+3a+\frac{9}{4}=\frac{17}{4}$$

Factor a^{2}+3a+\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}.

$$\left(a+\frac{3}{2}\right)^{2}=\frac{17}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{3}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Simplify.

$$a+\frac{3}{2}=\frac{\sqrt{17}}{2} a+\frac{3}{2}=-\frac{\sqrt{17}}{2}$$

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

$$a=\frac{\sqrt{17}-3}{2} a=\frac{-\sqrt{17}-3}{2}$$