Solve x+1/x+3-x/x+2=1/28

$$\frac { x + 1 } { x + 3 } - \frac { x } { x + 2 } = \frac { 1 } { 28 }$$

Solve for x

$$x=-10$$

$$x=5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,-2 since division by zero is not defined. Multiply both sides of the equation by 28\left(x+2\right)\left(x+3\right), the least common multiple of x+3,x+2,28.

$$\left(28x+56\right)\left(x+1\right)-\left(28x+84\right)x=\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply 28x+56 by x+1 and combine like terms.

$$28x^{2}+84x+56-\left(28x+84\right)x=\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply 28x+84 by x.

$$28x^{2}+84x+56-\left(28x^{2}+84x\right)=\left(x+2\right)\left(x+3\right)$$

To find the opposite of 28x^{2}+84x, find the opposite of each term.

$$28x^{2}+84x+56-28x^{2}-84x=\left(x+2\right)\left(x+3\right)$$

Combine 28x^{2} and -28x^{2} to get 0.

$$84x+56-84x=\left(x+2\right)\left(x+3\right)$$

Combine 84x and -84x to get 0.

$$56=\left(x+2\right)\left(x+3\right)$$

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

$$56=x^{2}+5x+6$$

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

$$x^{2}+5x+6=56$$

Subtract 56 from both sides.

$$x^{2}+5x+6-56=0$$

Subtract 56 from 6 to get -50.

$$x^{2}+5x-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-50\right)}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-50\right)}}{2}$$

Multiply -4 times -50.

$$x=\frac{-5±\sqrt{25+200}}{2}$$

Add 25 to 200.

$$x=\frac{-5±\sqrt{225}}{2}$$

Take the square root of 225.

$$x=\frac{-5±15}{2}$$

Now solve the equation x=\frac{-5±15}{2} when ± is plus. Add -5 to 15.

$$x=\frac{10}{2}$$

Divide 10 by 2.

$$x=5$$

Now solve the equation x=\frac{-5±15}{2} when ± is minus. Subtract 15 from -5.

$$x=-\frac{20}{2}$$

Divide -20 by 2.

$$x=-10$$

The equation is now solved.

$$x=5 x=-10$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,-2 since division by zero is not defined. Multiply both sides of the equation by 28\left(x+2\right)\left(x+3\right), the least common multiple of x+3,x+2,28.

$$\left(28x+56\right)\left(x+1\right)-\left(28x+84\right)x=\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply 28x+56 by x+1 and combine like terms.

$$28x^{2}+84x+56-\left(28x+84\right)x=\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply 28x+84 by x.

$$28x^{2}+84x+56-\left(28x^{2}+84x\right)=\left(x+2\right)\left(x+3\right)$$

To find the opposite of 28x^{2}+84x, find the opposite of each term.

$$28x^{2}+84x+56-28x^{2}-84x=\left(x+2\right)\left(x+3\right)$$

Combine 28x^{2} and -28x^{2} to get 0.

$$84x+56-84x=\left(x+2\right)\left(x+3\right)$$

Combine 84x and -84x to get 0.

$$56=\left(x+2\right)\left(x+3\right)$$

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

$$56=x^{2}+5x+6$$

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

$$x^{2}+5x+6=56$$

Subtract 6 from both sides.

$$x^{2}+5x=56-6$$

Subtract 6 from 56 to get 50.

$$x^{2}+5x=50$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=50+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=50+\frac{25}{4}$$

Add 50 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{225}{4}$$

Factor x^{2}+5x+\frac{25}{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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{15}{2} x+\frac{5}{2}=-\frac{15}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=5 x=-10$$