Solve x+1/x+4-x/x+3=1/10

$$\frac { x + 1 } { x + 4 } - \frac { x } { x + 3 } = \frac { 1 } { 10 }$$

Solve for x

$$x=-9$$

$$x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,-3 since division by zero is not defined. Multiply both sides of the equation by 10\left(x+3\right)\left(x+4\right), the least common multiple of x+4,x+3,10.

$$\left(10x+30\right)\left(x+1\right)-\left(10x+40\right)x=\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply 10x+30 by x+1 and combine like terms.

$$10x^{2}+40x+30-\left(10x+40\right)x=\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply 10x+40 by x.

$$10x^{2}+40x+30-\left(10x^{2}+40x\right)=\left(x+3\right)\left(x+4\right)$$

To find the opposite of 10x^{2}+40x, find the opposite of each term.

$$10x^{2}+40x+30-10x^{2}-40x=\left(x+3\right)\left(x+4\right)$$

Combine 10x^{2} and -10x^{2} to get 0.

$$40x+30-40x=\left(x+3\right)\left(x+4\right)$$

Combine 40x and -40x to get 0.

$$30=\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x+3 by x+4 and combine like terms.

$$30=x^{2}+7x+12$$

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

$$x^{2}+7x+12=30$$

Subtract 30 from both sides.

$$x^{2}+7x+12-30=0$$

Subtract 30 from 12 to get -18.

$$x^{2}+7x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-18\right)}}{2}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$x=\frac{-7±\sqrt{49+72}}{2}$$

Add 49 to 72.

$$x=\frac{-7±\sqrt{121}}{2}$$

Take the square root of 121.

$$x=\frac{-7±11}{2}$$

Now solve the equation x=\frac{-7±11}{2} when ± is plus. Add -7 to 11.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-7±11}{2} when ± is minus. Subtract 11 from -7.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=2 x=-9$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,-3 since division by zero is not defined. Multiply both sides of the equation by 10\left(x+3\right)\left(x+4\right), the least common multiple of x+4,x+3,10.

$$\left(10x+30\right)\left(x+1\right)-\left(10x+40\right)x=\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply 10x+30 by x+1 and combine like terms.

$$10x^{2}+40x+30-\left(10x+40\right)x=\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply 10x+40 by x.

$$10x^{2}+40x+30-\left(10x^{2}+40x\right)=\left(x+3\right)\left(x+4\right)$$

To find the opposite of 10x^{2}+40x, find the opposite of each term.

$$10x^{2}+40x+30-10x^{2}-40x=\left(x+3\right)\left(x+4\right)$$

Combine 10x^{2} and -10x^{2} to get 0.

$$40x+30-40x=\left(x+3\right)\left(x+4\right)$$

Combine 40x and -40x to get 0.

$$30=\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x+3 by x+4 and combine like terms.

$$30=x^{2}+7x+12$$

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

$$x^{2}+7x+12=30$$

Subtract 12 from both sides.

$$x^{2}+7x=30-12$$

Subtract 12 from 30 to get 18.

$$x^{2}+7x=18$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=18+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+7x+\frac{49}{4}=18+\frac{49}{4}$$

Add 18 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{121}{4}$$

Factor x^{2}+7x+\frac{49}{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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{11}{2} x+\frac{7}{2}=-\frac{11}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=2 x=-9$$