Solve x/x-7+7/x+2=22x-91/x^2-5x-14

$$\frac { x } { x - 7 } + \frac { 7 } { x + 2 } = \frac { 22 x - 91 } { x ^ { 2 } - 5 x - 14 }$$

Solve for x

$$x=6$$

Steps Using Factoring

Variable x cannot be equal to any of the values -2,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x+2\right), the least common multiple of x-7,x+2,x^{2}-5x-14.

$$\left(x+2\right)x+\left(x-7\right)\times 7=22x-91$$

Use the distributive property to multiply x+2 by x.

$$x^{2}+2x+\left(x-7\right)\times 7=22x-91$$

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

$$x^{2}+2x+7x-49=22x-91$$

Combine 2x and 7x to get 9x.

$$x^{2}+9x-49=22x-91$$

Subtract 22x from both sides.

$$x^{2}+9x-49-22x=-91$$

Combine 9x and -22x to get -13x.

$$x^{2}-13x-49=-91$$

Add 91 to both sides.

$$x^{2}-13x-49+91=0$$

Add -49 and 91 to get 42.

$$x^{2}-13x+42=0$$

To solve the equation, factor x^{2}-13x+42 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-13 ab=42$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 42.

$$-1,-42 -2,-21 -3,-14 -6,-7$$

Calculate the sum for each pair.

$$-1-42=-43 -2-21=-23 -3-14=-17 -6-7=-13$$

The solution is the pair that gives sum -13.

$$a=-7 b=-6$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-7\right)\left(x-6\right)$$

To find equation solutions, solve x-7=0 and x-6=0.

$$x=7 x=6$$

Variable x cannot be equal to 7.

$$x=6$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x+2\right), the least common multiple of x-7,x+2,x^{2}-5x-14.

$$\left(x+2\right)x+\left(x-7\right)\times 7=22x-91$$

Use the distributive property to multiply x+2 by x.

$$x^{2}+2x+\left(x-7\right)\times 7=22x-91$$

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

$$x^{2}+2x+7x-49=22x-91$$

Combine 2x and 7x to get 9x.

$$x^{2}+9x-49=22x-91$$

Subtract 22x from both sides.

$$x^{2}+9x-49-22x=-91$$

Combine 9x and -22x to get -13x.

$$x^{2}-13x-49=-91$$

Add 91 to both sides.

$$x^{2}-13x-49+91=0$$

Add -49 and 91 to get 42.

$$x^{2}-13x+42=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+42. To find a and b, set up a system to be solved.

$$a+b=-13 ab=1\times 42=42$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 42.

$$-1,-42 -2,-21 -3,-14 -6,-7$$

Calculate the sum for each pair.

$$-1-42=-43 -2-21=-23 -3-14=-17 -6-7=-13$$

The solution is the pair that gives sum -13.

$$a=-7 b=-6$$

Rewrite x^{2}-13x+42 as \left(x^{2}-7x\right)+\left(-6x+42\right).

$$\left(x^{2}-7x\right)+\left(-6x+42\right)$$

Factor out x in the first and -6 in the second group.

$$x\left(x-7\right)-6\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(x-6\right)$$

To find equation solutions, solve x-7=0 and x-6=0.

$$x=7 x=6$$

Variable x cannot be equal to 7.

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x+2\right), the least common multiple of x-7,x+2,x^{2}-5x-14.

$$\left(x+2\right)x+\left(x-7\right)\times 7=22x-91$$

Use the distributive property to multiply x+2 by x.

$$x^{2}+2x+\left(x-7\right)\times 7=22x-91$$

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

$$x^{2}+2x+7x-49=22x-91$$

Combine 2x and 7x to get 9x.

$$x^{2}+9x-49=22x-91$$

Subtract 22x from both sides.

$$x^{2}+9x-49-22x=-91$$

Combine 9x and -22x to get -13x.

$$x^{2}-13x-49=-91$$

Add 91 to both sides.

$$x^{2}-13x-49+91=0$$

Add -49 and 91 to get 42.

$$x^{2}-13x+42=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -13 for b, and 42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 42}}{2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 42}}{2}$$

Multiply -4 times 42.

$$x=\frac{-\left(-13\right)±\sqrt{169-168}}{2}$$

Add 169 to -168.

$$x=\frac{-\left(-13\right)±\sqrt{1}}{2}$$

Take the square root of 1.

$$x=\frac{-\left(-13\right)±1}{2}$$

The opposite of -13 is 13.

$$x=\frac{13±1}{2}$$

Now solve the equation x=\frac{13±1}{2} when ± is plus. Add 13 to 1.

$$x=\frac{14}{2}$$

Divide 14 by 2.

$$x=7$$

Now solve the equation x=\frac{13±1}{2} when ± is minus. Subtract 1 from 13.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

The equation is now solved.

$$x=7 x=6$$

Variable x cannot be equal to 7.

$$x=6$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x+2\right), the least common multiple of x-7,x+2,x^{2}-5x-14.

$$\left(x+2\right)x+\left(x-7\right)\times 7=22x-91$$

Use the distributive property to multiply x+2 by x.

$$x^{2}+2x+\left(x-7\right)\times 7=22x-91$$

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

$$x^{2}+2x+7x-49=22x-91$$

Combine 2x and 7x to get 9x.

$$x^{2}+9x-49=22x-91$$

Subtract 22x from both sides.

$$x^{2}+9x-49-22x=-91$$

Combine 9x and -22x to get -13x.

$$x^{2}-13x-49=-91$$

Add 49 to both sides.

$$x^{2}-13x=-91+49$$

Add -91 and 49 to get -42.

$$x^{2}-13x=-42$$

Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-42+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-13x+\frac{169}{4}=-42+\frac{169}{4}$$

Add -42 to \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{1}{4}$$

Factor x^{2}-13x+\frac{169}{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}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x-\frac{13}{2}=\frac{1}{2} x-\frac{13}{2}=-\frac{1}{2}$$

Add \frac{13}{2} to both sides of the equation.

$$x=7 x=6$$

Variable x cannot be equal to 7.

$$x=6$$