Solve 1/x+4-1/x-7=11/30,x

$$\frac { 1 } { x + 4 } - \frac { 1 } { x - 7 } = \frac { 11 } { 30 } , x$$

Solve for x

$$x=1$$

$$x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,7 since division by zero is not defined. Multiply both sides of the equation by 30\left(x-7\right)\left(x+4\right), the least common multiple of x+4,x-7,30.

$$30x-210-\left(30x+120\right)=11\left(x-7\right)\left(x+4\right)$$

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

$$30x-210-30x-120=11\left(x-7\right)\left(x+4\right)$$

Combine 30x and -30x to get 0.

$$-210-120=11\left(x-7\right)\left(x+4\right)$$

Subtract 120 from -210 to get -330.

$$-330=11\left(x-7\right)\left(x+4\right)$$

Use the distributive property to multiply 11 by x-7.

$$-330=\left(11x-77\right)\left(x+4\right)$$

Use the distributive property to multiply 11x-77 by x+4 and combine like terms.

$$-330=11x^{2}-33x-308$$

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

$$11x^{2}-33x-308=-330$$

Add 330 to both sides.

$$11x^{2}-33x-308+330=0$$

Add -308 and 330 to get 22.

$$11x^{2}-33x+22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 11 for a, -33 for b, and 22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 11\times 22}}{2\times 11}$$

Square -33.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\times 11\times 22}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-33\right)±\sqrt{1089-44\times 22}}{2\times 11}$$

Multiply -44 times 22.

$$x=\frac{-\left(-33\right)±\sqrt{1089-968}}{2\times 11}$$

Add 1089 to -968.

$$x=\frac{-\left(-33\right)±\sqrt{121}}{2\times 11}$$

Take the square root of 121.

$$x=\frac{-\left(-33\right)±11}{2\times 11}$$

The opposite of -33 is 33.

$$x=\frac{33±11}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{33±11}{22}$$

Now solve the equation x=\frac{33±11}{22} when ± is plus. Add 33 to 11.

$$x=\frac{44}{22}$$

Divide 44 by 22.

$$x=2$$

Now solve the equation x=\frac{33±11}{22} when ± is minus. Subtract 11 from 33.

$$x=\frac{22}{22}$$

Divide 22 by 22.

$$x=1$$

The equation is now solved.

$$x=2 x=1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,7 since division by zero is not defined. Multiply both sides of the equation by 30\left(x-7\right)\left(x+4\right), the least common multiple of x+4,x-7,30.

$$30x-210-\left(30x+120\right)=11\left(x-7\right)\left(x+4\right)$$

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

$$30x-210-30x-120=11\left(x-7\right)\left(x+4\right)$$

Combine 30x and -30x to get 0.

$$-210-120=11\left(x-7\right)\left(x+4\right)$$

Subtract 120 from -210 to get -330.

$$-330=11\left(x-7\right)\left(x+4\right)$$

Use the distributive property to multiply 11 by x-7.

$$-330=\left(11x-77\right)\left(x+4\right)$$

Use the distributive property to multiply 11x-77 by x+4 and combine like terms.

$$-330=11x^{2}-33x-308$$

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

$$11x^{2}-33x-308=-330$$

Add 308 to both sides.

$$11x^{2}-33x=-330+308$$

Add -330 and 308 to get -22.

$$11x^{2}-33x=-22$$

Divide both sides by 11.

$$\frac{11x^{2}-33x}{11}=-\frac{22}{11}$$

Dividing by 11 undoes the multiplication by 11.

$$x^{2}+\left(-\frac{33}{11}\right)x=-\frac{22}{11}$$

Divide -33 by 11.

$$x^{2}-3x=-\frac{22}{11}$$

Divide -22 by 11.

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

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.

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

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

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

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

$$x^{2}-3x+\frac{9}{4}=\frac{1}{4}$$

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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

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

Add \frac{3}{2} to both sides of the equation.

$$x=2 x=1$$