Solve -2/x-2+1/(x-2)^2=9/16

$$\frac { - 2 } { x - 2 } + \frac { 1 } { ( x - 2 ) ^ { 2 } } = \frac { 9 } { 16 }$$

Solve for x

$$x = \frac{22}{9} = 2\frac{4}{9} \approx 2.444444444$$

$$x=-2$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 16\left(x-2\right)^{2}, the least common multiple of x-2,\left(x-2\right)^{2},16.

$$\left(16x-32\right)\left(-2\right)+16=9\left(x-2\right)^{2}$$

Use the distributive property to multiply 16x-32 by -2.

$$-32x+64+16=9\left(x-2\right)^{2}$$

Add 64 and 16 to get 80.

$$-32x+80=9\left(x-2\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$-32x+80=9\left(x^{2}-4x+4\right)$$

Use the distributive property to multiply 9 by x^{2}-4x+4.

$$-32x+80=9x^{2}-36x+36$$

Subtract 9x^{2} from both sides.

$$-32x+80-9x^{2}=-36x+36$$

Add 36x to both sides.

$$-32x+80-9x^{2}+36x=36$$

Combine -32x and 36x to get 4x.

$$4x+80-9x^{2}=36$$

Subtract 36 from both sides.

$$4x+80-9x^{2}-36=0$$

Subtract 36 from 80 to get 44.

$$4x+44-9x^{2}=0$$

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

$$-9x^{2}+4x+44=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx+44. To find a and b, set up a system to be solved.

$$a+b=4 ab=-9\times 44=-396$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -396.

$$-1,396 -2,198 -3,132 -4,99 -6,66 -9,44 -11,36 -12,33 -18,22$$

Calculate the sum for each pair.

$$-1+396=395 -2+198=196 -3+132=129 -4+99=95 -6+66=60 -9+44=35 -11+36=25 -12+33=21 -18+22=4$$

The solution is the pair that gives sum 4.

$$a=22 b=-18$$

Rewrite -9x^{2}+4x+44 as \left(-9x^{2}+22x\right)+\left(-18x+44\right).

$$\left(-9x^{2}+22x\right)+\left(-18x+44\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(9x-22\right)-2\left(9x-22\right)$$

Factor out common term 9x-22 by using distributive property.

$$\left(9x-22\right)\left(-x-2\right)$$

To find equation solutions, solve 9x-22=0 and -x-2=0.

$$x=\frac{22}{9} x=-2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 16\left(x-2\right)^{2}, the least common multiple of x-2,\left(x-2\right)^{2},16.

$$\left(16x-32\right)\left(-2\right)+16=9\left(x-2\right)^{2}$$

Use the distributive property to multiply 16x-32 by -2.

$$-32x+64+16=9\left(x-2\right)^{2}$$

Add 64 and 16 to get 80.

$$-32x+80=9\left(x-2\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$-32x+80=9\left(x^{2}-4x+4\right)$$

Use the distributive property to multiply 9 by x^{2}-4x+4.

$$-32x+80=9x^{2}-36x+36$$

Subtract 9x^{2} from both sides.

$$-32x+80-9x^{2}=-36x+36$$

Add 36x to both sides.

$$-32x+80-9x^{2}+36x=36$$

Combine -32x and 36x to get 4x.

$$4x+80-9x^{2}=36$$

Subtract 36 from both sides.

$$4x+80-9x^{2}-36=0$$

Subtract 36 from 80 to get 44.

$$4x+44-9x^{2}=0$$

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.

$$-9x^{2}+4x+44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 4 for b, and 44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-9\right)\times 44}}{2\left(-9\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-9\right)\times 44}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-4±\sqrt{16+36\times 44}}{2\left(-9\right)}$$

Multiply 36 times 44.

$$x=\frac{-4±\sqrt{16+1584}}{2\left(-9\right)}$$

Add 16 to 1584.

$$x=\frac{-4±\sqrt{1600}}{2\left(-9\right)}$$

Take the square root of 1600.

$$x=\frac{-4±40}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-4±40}{-18}$$

Now solve the equation x=\frac{-4±40}{-18} when ± is plus. Add -4 to 40.

$$x=\frac{36}{-18}$$

Divide 36 by -18.

$$x=-2$$

Now solve the equation x=\frac{-4±40}{-18} when ± is minus. Subtract 40 from -4.

$$x=-\frac{44}{-18}$$

Reduce the fraction \frac{-44}{-18} to lowest terms by extracting and canceling out 2.

$$x=\frac{22}{9}$$

The equation is now solved.

$$x=-2 x=\frac{22}{9}$$

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 16\left(x-2\right)^{2}, the least common multiple of x-2,\left(x-2\right)^{2},16.

$$\left(16x-32\right)\left(-2\right)+16=9\left(x-2\right)^{2}$$

Use the distributive property to multiply 16x-32 by -2.

$$-32x+64+16=9\left(x-2\right)^{2}$$

Add 64 and 16 to get 80.

$$-32x+80=9\left(x-2\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$-32x+80=9\left(x^{2}-4x+4\right)$$

Use the distributive property to multiply 9 by x^{2}-4x+4.

$$-32x+80=9x^{2}-36x+36$$

Subtract 9x^{2} from both sides.

$$-32x+80-9x^{2}=-36x+36$$

Add 36x to both sides.

$$-32x+80-9x^{2}+36x=36$$

Combine -32x and 36x to get 4x.

$$4x+80-9x^{2}=36$$

Subtract 80 from both sides.

$$4x-9x^{2}=36-80$$

Subtract 80 from 36 to get -44.

$$4x-9x^{2}=-44$$

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.

$$-9x^{2}+4x=-44$$

Divide both sides by -9.

$$\frac{-9x^{2}+4x}{-9}=-\frac{44}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{4}{-9}x=-\frac{44}{-9}$$

Divide 4 by -9.

$$x^{2}-\frac{4}{9}x=-\frac{44}{-9}$$

Divide -44 by -9.

$$x^{2}-\frac{4}{9}x=\frac{44}{9}$$

Divide -\frac{4}{9}, the coefficient of the x term, by 2 to get -\frac{2}{9}. Then add the square of -\frac{2}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{9}x+\left(-\frac{2}{9}\right)^{2}=\frac{44}{9}+\left(-\frac{2}{9}\right)^{2}$$

Square -\frac{2}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{9}x+\frac{4}{81}=\frac{44}{9}+\frac{4}{81}$$

Add \frac{44}{9} to \frac{4}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{9}x+\frac{4}{81}=\frac{400}{81}$$

Factor x^{2}-\frac{4}{9}x+\frac{4}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{9}\right)^{2}=\frac{400}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{9}\right)^{2}}=\sqrt{\frac{400}{81}}$$

Simplify.

$$x-\frac{2}{9}=\frac{20}{9} x-\frac{2}{9}=-\frac{20}{9}$$

Add \frac{2}{9} to both sides of the equation.

$$x=\frac{22}{9} x=-2$$