Solve 3/x-4+x-3/x=2

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

Solve for x

$$x=-2$$

$$x=6$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by x\left(x-4\right), the least common multiple of x-4,x.

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

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$x\times 3+x^{2}-7x+12=2x\left(x-4\right)$$

Combine x\times 3 and -7x to get -4x.

$$-4x+x^{2}+12=2x\left(x-4\right)$$

Use the distributive property to multiply 2x by x-4.

$$-4x+x^{2}+12=2x^{2}-8x$$

Subtract 2x^{2} from both sides.

$$-4x+x^{2}+12-2x^{2}=-8x$$

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

$$-4x-x^{2}+12=-8x$$

Add 8x to both sides.

$$-4x-x^{2}+12+8x=0$$

Combine -4x and 8x to get 4x.

$$4x-x^{2}+12=0$$

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

$$-x^{2}+4x+12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=4 ab=-12=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 4.

$$a=6 b=-2$$

Rewrite -x^{2}+4x+12 as \left(-x^{2}+6x\right)+\left(-2x+12\right).

$$\left(-x^{2}+6x\right)+\left(-2x+12\right)$$

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

$$-x\left(x-6\right)-2\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(-x-2\right)$$

To find equation solutions, solve x-6=0 and -x-2=0.

$$x=6 x=-2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by x\left(x-4\right), the least common multiple of x-4,x.

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

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$x\times 3+x^{2}-7x+12=2x\left(x-4\right)$$

Combine x\times 3 and -7x to get -4x.

$$-4x+x^{2}+12=2x\left(x-4\right)$$

Use the distributive property to multiply 2x by x-4.

$$-4x+x^{2}+12=2x^{2}-8x$$

Subtract 2x^{2} from both sides.

$$-4x+x^{2}+12-2x^{2}=-8x$$

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

$$-4x-x^{2}+12=-8x$$

Add 8x to both sides.

$$-4x-x^{2}+12+8x=0$$

Combine -4x and 8x to get 4x.

$$4x-x^{2}+12=0$$

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.

$$-x^{2}+4x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\times 12}}{2\left(-1\right)}$$

Multiply 4 times 12.

$$x=\frac{-4±\sqrt{16+48}}{2\left(-1\right)}$$

Add 16 to 48.

$$x=\frac{-4±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$x=\frac{-4±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±8}{-2}$$

Now solve the equation x=\frac{-4±8}{-2} when ± is plus. Add -4 to 8.

$$x=\frac{4}{-2}$$

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-4±8}{-2} when ± is minus. Subtract 8 from -4.

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

Divide -12 by -2.

$$x=6$$

The equation is now solved.

$$x=-2 x=6$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by x\left(x-4\right), the least common multiple of x-4,x.

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

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$x\times 3+x^{2}-7x+12=2x\left(x-4\right)$$

Combine x\times 3 and -7x to get -4x.

$$-4x+x^{2}+12=2x\left(x-4\right)$$

Use the distributive property to multiply 2x by x-4.

$$-4x+x^{2}+12=2x^{2}-8x$$

Subtract 2x^{2} from both sides.

$$-4x+x^{2}+12-2x^{2}=-8x$$

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

$$-4x-x^{2}+12=-8x$$

Add 8x to both sides.

$$-4x-x^{2}+12+8x=0$$

Combine -4x and 8x to get 4x.

$$4x-x^{2}+12=0$$

Subtract 12 from both sides. Anything subtracted from zero gives its negation.

$$4x-x^{2}=-12$$

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.

$$-x^{2}+4x=-12$$

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=-\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=-\frac{12}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=-\frac{12}{-1}$$

Divide -12 by -1.

$$x^{2}-4x=12$$

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

$$x^{2}-4x+\left(-2\right)^{2}=12+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=12+4$$

Add 12 to 4.

$$x^{2}-4x+4=16$$

Factor x^{2}-4x+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}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-2=4 x-2=-4$$

Add 2 to both sides of the equation.

$$x=6 x=-2$$