Solve 1/x+4-1/x+1=11/30

$$\frac { 1 } { x + 4 } - \frac { 1 } { x + 1 } = \frac { 11 } { 30 }$$

Solve for x (complex solution)

$$x=-\frac{3\sqrt{319}i}{22}-\frac{5}{2}\approx -2.5-2.435532423i$$

$$x=\frac{3\sqrt{319}i}{22}-\frac{5}{2}\approx -2.5+2.435532423i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,-1 since division by zero is not defined. Multiply both sides of the equation by 30\left(x+1\right)\left(x+4\right), the least common multiple of x+4,x+1,30.

$$30x+30-\left(30x+120\right)=11\left(x+1\right)\left(x+4\right)$$

To find the opposite of 30x+120, find the opposite of each term.

$$30x+30-30x-120=11\left(x+1\right)\left(x+4\right)$$

Combine 30x and -30x to get 0.

$$30-120=11\left(x+1\right)\left(x+4\right)$$

Subtract 120 from 30 to get -90.

$$-90=11\left(x+1\right)\left(x+4\right)$$

Use the distributive property to multiply 11 by x+1.

$$-90=\left(11x+11\right)\left(x+4\right)$$

Use the distributive property to multiply 11x+11 by x+4 and combine like terms.

$$-90=11x^{2}+55x+44$$

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

$$11x^{2}+55x+44=-90$$

Add 90 to both sides.

$$11x^{2}+55x+44+90=0$$

Add 44 and 90 to get 134.

$$11x^{2}+55x+134=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 11 for a, 55 for b, and 134 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-55±\sqrt{55^{2}-4\times 11\times 134}}{2\times 11}$$

Square 55.

$$x=\frac{-55±\sqrt{3025-4\times 11\times 134}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-55±\sqrt{3025-44\times 134}}{2\times 11}$$

Multiply -44 times 134.

$$x=\frac{-55±\sqrt{3025-5896}}{2\times 11}$$

Add 3025 to -5896.

$$x=\frac{-55±\sqrt{-2871}}{2\times 11}$$

Take the square root of -2871.

$$x=\frac{-55±3\sqrt{319}i}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{-55±3\sqrt{319}i}{22}$$

Now solve the equation x=\frac{-55±3\sqrt{319}i}{22} when ± is plus. Add -55 to 3i\sqrt{319}.

$$x=\frac{-55+3\sqrt{319}i}{22}$$

Divide -55+3i\sqrt{319} by 22.

$$x=\frac{3\sqrt{319}i}{22}-\frac{5}{2}$$

Now solve the equation x=\frac{-55±3\sqrt{319}i}{22} when ± is minus. Subtract 3i\sqrt{319} from -55.

$$x=\frac{-3\sqrt{319}i-55}{22}$$

Divide -55-3i\sqrt{319} by 22.

$$x=-\frac{3\sqrt{319}i}{22}-\frac{5}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{319}i}{22}-\frac{5}{2} x=-\frac{3\sqrt{319}i}{22}-\frac{5}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,-1 since division by zero is not defined. Multiply both sides of the equation by 30\left(x+1\right)\left(x+4\right), the least common multiple of x+4,x+1,30.

$$30x+30-\left(30x+120\right)=11\left(x+1\right)\left(x+4\right)$$

To find the opposite of 30x+120, find the opposite of each term.

$$30x+30-30x-120=11\left(x+1\right)\left(x+4\right)$$

Combine 30x and -30x to get 0.

$$30-120=11\left(x+1\right)\left(x+4\right)$$

Subtract 120 from 30 to get -90.

$$-90=11\left(x+1\right)\left(x+4\right)$$

Use the distributive property to multiply 11 by x+1.

$$-90=\left(11x+11\right)\left(x+4\right)$$

Use the distributive property to multiply 11x+11 by x+4 and combine like terms.

$$-90=11x^{2}+55x+44$$

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

$$11x^{2}+55x+44=-90$$

Subtract 44 from both sides.

$$11x^{2}+55x=-90-44$$

Subtract 44 from -90 to get -134.

$$11x^{2}+55x=-134$$

Divide both sides by 11.

$$\frac{11x^{2}+55x}{11}=-\frac{134}{11}$$

Dividing by 11 undoes the multiplication by 11.

$$x^{2}+\frac{55}{11}x=-\frac{134}{11}$$

Divide 55 by 11.

$$x^{2}+5x=-\frac{134}{11}$$

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.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-\frac{134}{11}+\left(\frac{5}{2}\right)^{2}$$

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

$$x^{2}+5x+\frac{25}{4}=-\frac{134}{11}+\frac{25}{4}$$

Add -\frac{134}{11} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+5x+\frac{25}{4}=-\frac{261}{44}$$

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}.

$$\left(x+\frac{5}{2}\right)^{2}=-\frac{261}{44}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{-\frac{261}{44}}$$

Simplify.

$$x+\frac{5}{2}=\frac{3\sqrt{319}i}{22} x+\frac{5}{2}=-\frac{3\sqrt{319}i}{22}$$

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

$$x=\frac{3\sqrt{319}i}{22}-\frac{5}{2} x=-\frac{3\sqrt{319}i}{22}-\frac{5}{2}$$