Solve 522/x-609/x+11=8

$$\frac{ 522 }{ x } - \frac{ 609 }{ x+11 } = 8$$

Solve for x

$$x = -\frac{319}{8} = -39\frac{7}{8} = -39.875$$

$$x=18$$

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.

$$\left(x+11\right)\times 522-x\times 609=8x\left(x+11\right)$$

Use the distributive property to multiply x+11 by 522.

$$522x+5742-x\times 609=8x\left(x+11\right)$$

Use the distributive property to multiply 8x by x+11.

$$522x+5742-x\times 609=8x^{2}+88x$$

Subtract 8x^{2} from both sides.

$$522x+5742-x\times 609-8x^{2}=88x$$

Subtract 88x from both sides.

$$522x+5742-x\times 609-8x^{2}-88x=0$$

Combine 522x and -88x to get 434x.

$$434x+5742-x\times 609-8x^{2}=0$$

Multiply -1 and 609 to get -609.

$$434x+5742-609x-8x^{2}=0$$

Combine 434x and -609x to get -175x.

$$-175x+5742-8x^{2}=0$$

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

$$-8x^{2}-175x+5742=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -8x^{2}+ax+bx+5742. To find a and b, set up a system to be solved.

$$a+b=-175 ab=-8\times 5742=-45936$$

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 -45936.

$$1,-45936 2,-22968 3,-15312 4,-11484 6,-7656 8,-5742 9,-5104 11,-4176 12,-3828 16,-2871 18,-2552 22,-2088 24,-1914 29,-1584 33,-1392 36,-1276 44,-1044 48,-957 58,-792 66,-696 72,-638 87,-528 88,-522 99,-464 116,-396 132,-348 144,-319 174,-264 176,-261 198,-232$$

Calculate the sum for each pair.

$$1-45936=-45935 2-22968=-22966 3-15312=-15309 4-11484=-11480 6-7656=-7650 8-5742=-5734 9-5104=-5095 11-4176=-4165 12-3828=-3816 16-2871=-2855 18-2552=-2534 22-2088=-2066 24-1914=-1890 29-1584=-1555 33-1392=-1359 36-1276=-1240 44-1044=-1000 48-957=-909 58-792=-734 66-696=-630 72-638=-566 87-528=-441 88-522=-434 99-464=-365 116-396=-280 132-348=-216 144-319=-175 174-264=-90 176-261=-85 198-232=-34$$

The solution is the pair that gives sum -175.

$$a=144 b=-319$$

Rewrite -8x^{2}-175x+5742 as \left(-8x^{2}+144x\right)+\left(-319x+5742\right).

$$\left(-8x^{2}+144x\right)+\left(-319x+5742\right)$$

Factor out 8x in the first and 319 in the second group.

$$8x\left(-x+18\right)+319\left(-x+18\right)$$

Factor out common term -x+18 by using distributive property.

$$\left(-x+18\right)\left(8x+319\right)$$

To find equation solutions, solve -x+18=0 and 8x+319=0.

$$x=18 x=-\frac{319}{8}$$

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.

$$\left(x+11\right)\times 522-x\times 609=8x\left(x+11\right)$$

Use the distributive property to multiply x+11 by 522.

$$522x+5742-x\times 609=8x\left(x+11\right)$$

Use the distributive property to multiply 8x by x+11.

$$522x+5742-x\times 609=8x^{2}+88x$$

Subtract 8x^{2} from both sides.

$$522x+5742-x\times 609-8x^{2}=88x$$

Subtract 88x from both sides.

$$522x+5742-x\times 609-8x^{2}-88x=0$$

Combine 522x and -88x to get 434x.

$$434x+5742-x\times 609-8x^{2}=0$$

Multiply -1 and 609 to get -609.

$$434x+5742-609x-8x^{2}=0$$

Combine 434x and -609x to get -175x.

$$-175x+5742-8x^{2}=0$$

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.

$$-8x^{2}-175x+5742=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -175 for b, and 5742 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-175\right)±\sqrt{\left(-175\right)^{2}-4\left(-8\right)\times 5742}}{2\left(-8\right)}$$

Square -175.

$$x=\frac{-\left(-175\right)±\sqrt{30625-4\left(-8\right)\times 5742}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-\left(-175\right)±\sqrt{30625+32\times 5742}}{2\left(-8\right)}$$

Multiply 32 times 5742.

$$x=\frac{-\left(-175\right)±\sqrt{30625+183744}}{2\left(-8\right)}$$

Add 30625 to 183744.

$$x=\frac{-\left(-175\right)±\sqrt{214369}}{2\left(-8\right)}$$

Take the square root of 214369.

$$x=\frac{-\left(-175\right)±463}{2\left(-8\right)}$$

The opposite of -175 is 175.

$$x=\frac{175±463}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{175±463}{-16}$$

Now solve the equation x=\frac{175±463}{-16} when ± is plus. Add 175 to 463.

$$x=\frac{638}{-16}$$

Reduce the fraction \frac{638}{-16} to lowest terms by extracting and canceling out 2.

$$x=-\frac{319}{8}$$

Now solve the equation x=\frac{175±463}{-16} when ± is minus. Subtract 463 from 175.

$$x=-\frac{288}{-16}$$

Divide -288 by -16.

$$x=18$$

The equation is now solved.

$$x=-\frac{319}{8} x=18$$

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.

$$\left(x+11\right)\times 522-x\times 609=8x\left(x+11\right)$$

Use the distributive property to multiply x+11 by 522.

$$522x+5742-x\times 609=8x\left(x+11\right)$$

Use the distributive property to multiply 8x by x+11.

$$522x+5742-x\times 609=8x^{2}+88x$$

Subtract 8x^{2} from both sides.

$$522x+5742-x\times 609-8x^{2}=88x$$

Subtract 88x from both sides.

$$522x+5742-x\times 609-8x^{2}-88x=0$$

Combine 522x and -88x to get 434x.

$$434x+5742-x\times 609-8x^{2}=0$$

Subtract 5742 from both sides. Anything subtracted from zero gives its negation.

$$434x-x\times 609-8x^{2}=-5742$$

Multiply -1 and 609 to get -609.

$$434x-609x-8x^{2}=-5742$$

Combine 434x and -609x to get -175x.

$$-175x-8x^{2}=-5742$$

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.

$$-8x^{2}-175x=-5742$$

Divide both sides by -8.

$$\frac{-8x^{2}-175x}{-8}=-\frac{5742}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\left(-\frac{175}{-8}\right)x=-\frac{5742}{-8}$$

Divide -175 by -8.

$$x^{2}+\frac{175}{8}x=-\frac{5742}{-8}$$

Reduce the fraction \frac{-5742}{-8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{175}{8}x=\frac{2871}{4}$$

Divide \frac{175}{8}, the coefficient of the x term, by 2 to get \frac{175}{16}. Then add the square of \frac{175}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{175}{8}x+\left(\frac{175}{16}\right)^{2}=\frac{2871}{4}+\left(\frac{175}{16}\right)^{2}$$

Square \frac{175}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{175}{8}x+\frac{30625}{256}=\frac{2871}{4}+\frac{30625}{256}$$

Add \frac{2871}{4} to \frac{30625}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{175}{8}x+\frac{30625}{256}=\frac{214369}{256}$$

Factor x^{2}+\frac{175}{8}x+\frac{30625}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{175}{16}\right)^{2}=\frac{214369}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{175}{16}\right)^{2}}=\sqrt{\frac{214369}{256}}$$

Simplify.

$$x+\frac{175}{16}=\frac{463}{16} x+\frac{175}{16}=-\frac{463}{16}$$

Subtract \frac{175}{16} from both sides of the equation.

$$x=18 x=-\frac{319}{8}$$