Solve 1/x-1/x+2=-4/3(x-3)

$$\frac { 1 } { x } - \frac { 1 } { x + 2 } = - \frac { 4 } { 3 ( x - 3 ) }$$

Solve for x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$x=1$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,0,3 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x-3\right)\left(x+2\right), the least common multiple of x,x+2,3\left(x-3\right).

$$3\left(x-3\right)\left(x+2\right)-3x\left(x-3\right)=-x\left(x+2\right)\times 4$$

Use the distributive property to multiply 3 by x-3.

$$\left(3x-9\right)\left(x+2\right)-3x\left(x-3\right)=-x\left(x+2\right)\times 4$$

Use the distributive property to multiply 3x-9 by x+2 and combine like terms.

$$3x^{2}-3x-18-3x\left(x-3\right)=-x\left(x+2\right)\times 4$$

Use the distributive property to multiply -3x by x-3.

$$3x^{2}-3x-18-3x^{2}+9x=-x\left(x+2\right)\times 4$$

Combine 3x^{2} and -3x^{2} to get 0.

$$-3x-18+9x=-x\left(x+2\right)\times 4$$

Combine -3x and 9x to get 6x.

$$6x-18=-x\left(x+2\right)\times 4$$

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

$$6x-18=-\left(x^{2}+2x\right)\times 4$$

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

$$6x-18=-\left(4x^{2}+8x\right)$$

To find the opposite of 4x^{2}+8x, find the opposite of each term.

$$6x-18=-4x^{2}-8x$$

Add 4x^{2} to both sides.

$$6x-18+4x^{2}=-8x$$

Add 8x to both sides.

$$6x-18+4x^{2}+8x=0$$

Combine 6x and 8x to get 14x.

$$14x-18+4x^{2}=0$$

Divide both sides by 2.

$$7x-9+2x^{2}=0$$

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

$$2x^{2}+7x-9=0$$

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

$$a+b=7 ab=2\left(-9\right)=-18$$

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 -18.

$$-1,18 -2,9 -3,6$$

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 7.

$$a=-2 b=9$$

Rewrite 2x^{2}+7x-9 as \left(2x^{2}-2x\right)+\left(9x-9\right).

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

Factor out 2x in the first and 9 in the second group.

$$2x\left(x-1\right)+9\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(2x+9\right)$$

To find equation solutions, solve x-1=0 and 2x+9=0.

$$x=1 x=-\frac{9}{2}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0,3 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x-3\right)\left(x+2\right), the least common multiple of x,x+2,3\left(x-3\right).

$$3\left(x-3\right)\left(x+2\right)-3x\left(x-3\right)=-x\left(x+2\right)\times 4$$

Use the distributive property to multiply 3 by x-3.

$$\left(3x-9\right)\left(x+2\right)-3x\left(x-3\right)=-x\left(x+2\right)\times 4$$

Use the distributive property to multiply 3x-9 by x+2 and combine like terms.

$$3x^{2}-3x-18-3x\left(x-3\right)=-x\left(x+2\right)\times 4$$

Use the distributive property to multiply -3x by x-3.

$$3x^{2}-3x-18-3x^{2}+9x=-x\left(x+2\right)\times 4$$

Combine 3x^{2} and -3x^{2} to get 0.

$$-3x-18+9x=-x\left(x+2\right)\times 4$$

Combine -3x and 9x to get 6x.

$$6x-18=-x\left(x+2\right)\times 4$$

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

$$6x-18=-\left(x^{2}+2x\right)\times 4$$

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

$$6x-18=-\left(4x^{2}+8x\right)$$

To find the opposite of 4x^{2}+8x, find the opposite of each term.

$$6x-18=-4x^{2}-8x$$

Add 4x^{2} to both sides.

$$6x-18+4x^{2}=-8x$$

Add 8x to both sides.

$$6x-18+4x^{2}+8x=0$$

Combine 6x and 8x to get 14x.

$$14x-18+4x^{2}=0$$

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.

$$4x^{2}+14x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 14 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 4\left(-18\right)}}{2\times 4}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 4\left(-18\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-14±\sqrt{196-16\left(-18\right)}}{2\times 4}$$

Multiply -16 times -18.

$$x=\frac{-14±\sqrt{196+288}}{2\times 4}$$

Add 196 to 288.

$$x=\frac{-14±\sqrt{484}}{2\times 4}$$

Take the square root of 484.

$$x=\frac{-14±22}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-14±22}{8}$$

Now solve the equation x=\frac{-14±22}{8} when ± is plus. Add -14 to 22.

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

Divide 8 by 8.

$$x=1$$

Now solve the equation x=\frac{-14±22}{8} when ± is minus. Subtract 22 from -14.

$$x=-\frac{36}{8}$$

Reduce the fraction \frac{-36}{8} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

$$x=1 x=-\frac{9}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0,3 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x-3\right)\left(x+2\right), the least common multiple of x,x+2,3\left(x-3\right).

$$3\left(x-3\right)\left(x+2\right)-3x\left(x-3\right)=-x\left(x+2\right)\times 4$$

Use the distributive property to multiply 3 by x-3.

$$\left(3x-9\right)\left(x+2\right)-3x\left(x-3\right)=-x\left(x+2\right)\times 4$$

Use the distributive property to multiply 3x-9 by x+2 and combine like terms.

$$3x^{2}-3x-18-3x\left(x-3\right)=-x\left(x+2\right)\times 4$$

Use the distributive property to multiply -3x by x-3.

$$3x^{2}-3x-18-3x^{2}+9x=-x\left(x+2\right)\times 4$$

Combine 3x^{2} and -3x^{2} to get 0.

$$-3x-18+9x=-x\left(x+2\right)\times 4$$

Combine -3x and 9x to get 6x.

$$6x-18=-x\left(x+2\right)\times 4$$

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

$$6x-18=-\left(x^{2}+2x\right)\times 4$$

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

$$6x-18=-\left(4x^{2}+8x\right)$$

To find the opposite of 4x^{2}+8x, find the opposite of each term.

$$6x-18=-4x^{2}-8x$$

Add 4x^{2} to both sides.

$$6x-18+4x^{2}=-8x$$

Add 8x to both sides.

$$6x-18+4x^{2}+8x=0$$

Combine 6x and 8x to get 14x.

$$14x-18+4x^{2}=0$$

Add 18 to both sides. Anything plus zero gives itself.

$$14x+4x^{2}=18$$

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.

$$4x^{2}+14x=18$$

Divide both sides by 4.

$$\frac{4x^{2}+14x}{4}=\frac{18}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{14}{4}x=\frac{18}{4}$$

Reduce the fraction \frac{14}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{7}{2}x=\frac{18}{4}$$

Reduce the fraction \frac{18}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{7}{2}x=\frac{9}{2}$$

Divide \frac{7}{2}, the coefficient of the x term, by 2 to get \frac{7}{4}. Then add the square of \frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=\frac{9}{2}+\left(\frac{7}{4}\right)^{2}$$

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{9}{2}+\frac{49}{16}$$

Add \frac{9}{2} to \frac{49}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{121}{16}$$

Factor x^{2}+\frac{7}{2}x+\frac{49}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{121}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{121}{16}}$$

Simplify.

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

Subtract \frac{7}{4} from both sides of the equation.

$$x=1 x=-\frac{9}{2}$$