Solve 132/x-132/x+11=1

$$\frac { 132 } { x } - \frac { 132 } { x + 11 } = 1$$

Solve for x

$$x=-44$$

$$x=33$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -11,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+11\right), the least common multiple of x,x+11.

$$\left(x+11\right)\times 132-x\times 132=x\left(x+11\right)$$

Use the distributive property to multiply x+11 by 132.

$$132x+1452-x\times 132=x\left(x+11\right)$$

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

$$132x+1452-x\times 132=x^{2}+11x$$

Subtract x^{2} from both sides.

$$132x+1452-x\times 132-x^{2}=11x$$

Subtract 11x from both sides.

$$132x+1452-x\times 132-x^{2}-11x=0$$

Combine 132x and -11x to get 121x.

$$121x+1452-x\times 132-x^{2}=0$$

Multiply -1 and 132 to get -132.

$$121x+1452-132x-x^{2}=0$$

Combine 121x and -132x to get -11x.

$$-11x+1452-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-11x+1452=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+1452. To find a and b, set up a system to be solved.

$$a+b=-11 ab=-1452=-1452$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1452.

$$1,-1452 2,-726 3,-484 4,-363 6,-242 11,-132 12,-121 22,-66 33,-44$$

Calculate the sum for each pair.

$$1-1452=-1451 2-726=-724 3-484=-481 4-363=-359 6-242=-236 11-132=-121 12-121=-109 22-66=-44 33-44=-11$$

The solution is the pair that gives sum -11.

$$a=33 b=-44$$

Rewrite -x^{2}-11x+1452 as \left(-x^{2}+33x\right)+\left(-44x+1452\right).

$$\left(-x^{2}+33x\right)+\left(-44x+1452\right)$$

Factor out x in the first and 44 in the second group.

$$x\left(-x+33\right)+44\left(-x+33\right)$$

Factor out common term -x+33 by using distributive property.

$$\left(-x+33\right)\left(x+44\right)$$

To find equation solutions, solve -x+33=0 and x+44=0.

$$x=33 x=-44$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -11,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+11\right), the least common multiple of x,x+11.

$$\left(x+11\right)\times 132-x\times 132=x\left(x+11\right)$$

Use the distributive property to multiply x+11 by 132.

$$132x+1452-x\times 132=x\left(x+11\right)$$

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

$$132x+1452-x\times 132=x^{2}+11x$$

Subtract x^{2} from both sides.

$$132x+1452-x\times 132-x^{2}=11x$$

Subtract 11x from both sides.

$$132x+1452-x\times 132-x^{2}-11x=0$$

Combine 132x and -11x to get 121x.

$$121x+1452-x\times 132-x^{2}=0$$

Multiply -1 and 132 to get -132.

$$121x+1452-132x-x^{2}=0$$

Combine 121x and -132x to get -11x.

$$-11x+1452-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}-11x+1452=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -11 for b, and 1452 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-1\right)\times 1452}}{2\left(-1\right)}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-1\right)\times 1452}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-11\right)±\sqrt{121+4\times 1452}}{2\left(-1\right)}$$

Multiply 4 times 1452.

$$x=\frac{-\left(-11\right)±\sqrt{121+5808}}{2\left(-1\right)}$$

Add 121 to 5808.

$$x=\frac{-\left(-11\right)±\sqrt{5929}}{2\left(-1\right)}$$

Take the square root of 5929.

$$x=\frac{-\left(-11\right)±77}{2\left(-1\right)}$$

The opposite of -11 is 11.

$$x=\frac{11±77}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{11±77}{-2}$$

Now solve the equation x=\frac{11±77}{-2} when ± is plus. Add 11 to 77.

$$x=\frac{88}{-2}$$

Divide 88 by -2.

$$x=-44$$

Now solve the equation x=\frac{11±77}{-2} when ± is minus. Subtract 77 from 11.

$$x=-\frac{66}{-2}$$

Divide -66 by -2.

$$x=33$$

The equation is now solved.

$$x=-44 x=33$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -11,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+11\right), the least common multiple of x,x+11.

$$\left(x+11\right)\times 132-x\times 132=x\left(x+11\right)$$

Use the distributive property to multiply x+11 by 132.

$$132x+1452-x\times 132=x\left(x+11\right)$$

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

$$132x+1452-x\times 132=x^{2}+11x$$

Subtract x^{2} from both sides.

$$132x+1452-x\times 132-x^{2}=11x$$

Subtract 11x from both sides.

$$132x+1452-x\times 132-x^{2}-11x=0$$

Combine 132x and -11x to get 121x.

$$121x+1452-x\times 132-x^{2}=0$$

Subtract 1452 from both sides. Anything subtracted from zero gives its negation.

$$121x-x\times 132-x^{2}=-1452$$

Multiply -1 and 132 to get -132.

$$121x-132x-x^{2}=-1452$$

Combine 121x and -132x to get -11x.

$$-11x-x^{2}=-1452$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}-11x=-1452$$

Divide both sides by -1.

$$\frac{-x^{2}-11x}{-1}=-\frac{1452}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{11}{-1}\right)x=-\frac{1452}{-1}$$

Divide -11 by -1.

$$x^{2}+11x=-\frac{1452}{-1}$$

Divide -1452 by -1.

$$x^{2}+11x=1452$$

Divide 11, the coefficient of the x term, by 2 to get \frac{11}{2}. Then add the square of \frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=1452+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11x+\frac{121}{4}=1452+\frac{121}{4}$$

Add 1452 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{5929}{4}$$

Factor x^{2}+11x+\frac{121}{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}.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{5929}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{5929}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{77}{2} x+\frac{11}{2}=-\frac{77}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=33 x=-44$$