Solve 6/x+3/x-1=6

$$\frac { 6 } { x } + \frac { 3 } { x - 1 } = 6$$

Solve for x

$$x=2$$

$$x=\frac{1}{2}=0.5$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x,x-1.

$$\left(x-1\right)\times 6+x\times 3=6x\left(x-1\right)$$

Use the distributive property to multiply x-1 by 6.

$$6x-6+x\times 3=6x\left(x-1\right)$$

Combine 6x and x\times 3 to get 9x.

$$9x-6=6x\left(x-1\right)$$

Use the distributive property to multiply 6x by x-1.

$$9x-6=6x^{2}-6x$$

Subtract 6x^{2} from both sides.

$$9x-6-6x^{2}=-6x$$

Add 6x to both sides.

$$9x-6-6x^{2}+6x=0$$

Combine 9x and 6x to get 15x.

$$15x-6-6x^{2}=0$$

Divide both sides by 3.

$$5x-2-2x^{2}=0$$

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

$$-2x^{2}+5x-2=0$$

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

$$a+b=5 ab=-2\left(-2\right)=4$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=4 b=1$$

Rewrite -2x^{2}+5x-2 as \left(-2x^{2}+4x\right)+\left(x-2\right).

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

Factor out 2x in the first and -1 in the second group.

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

Factor out common term -x+2 by using distributive property.

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

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

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

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x,x-1.

$$\left(x-1\right)\times 6+x\times 3=6x\left(x-1\right)$$

Use the distributive property to multiply x-1 by 6.

$$6x-6+x\times 3=6x\left(x-1\right)$$

Combine 6x and x\times 3 to get 9x.

$$9x-6=6x\left(x-1\right)$$

Use the distributive property to multiply 6x by x-1.

$$9x-6=6x^{2}-6x$$

Subtract 6x^{2} from both sides.

$$9x-6-6x^{2}=-6x$$

Add 6x to both sides.

$$9x-6-6x^{2}+6x=0$$

Combine 9x and 6x to get 15x.

$$15x-6-6x^{2}=0$$

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.

$$-6x^{2}+15x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 15 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-6\right)\left(-6\right)}}{2\left(-6\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-6\right)\left(-6\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-15±\sqrt{225+24\left(-6\right)}}{2\left(-6\right)}$$

Multiply 24 times -6.

$$x=\frac{-15±\sqrt{225-144}}{2\left(-6\right)}$$

Add 225 to -144.

$$x=\frac{-15±\sqrt{81}}{2\left(-6\right)}$$

Take the square root of 81.

$$x=\frac{-15±9}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-15±9}{-12}$$

Now solve the equation x=\frac{-15±9}{-12} when ± is plus. Add -15 to 9.

$$x=-\frac{6}{-12}$$

Reduce the fraction \frac{-6}{-12} to lowest terms by extracting and canceling out 6.

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

Now solve the equation x=\frac{-15±9}{-12} when ± is minus. Subtract 9 from -15.

$$x=-\frac{24}{-12}$$

Divide -24 by -12.

$$x=2$$

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x,x-1.

$$\left(x-1\right)\times 6+x\times 3=6x\left(x-1\right)$$

Use the distributive property to multiply x-1 by 6.

$$6x-6+x\times 3=6x\left(x-1\right)$$

Combine 6x and x\times 3 to get 9x.

$$9x-6=6x\left(x-1\right)$$

Use the distributive property to multiply 6x by x-1.

$$9x-6=6x^{2}-6x$$

Subtract 6x^{2} from both sides.

$$9x-6-6x^{2}=-6x$$

Add 6x to both sides.

$$9x-6-6x^{2}+6x=0$$

Combine 9x and 6x to get 15x.

$$15x-6-6x^{2}=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$15x-6x^{2}=6$$

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.

$$-6x^{2}+15x=6$$

Divide both sides by -6.

$$\frac{-6x^{2}+15x}{-6}=\frac{6}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{15}{-6}x=\frac{6}{-6}$$

Reduce the fraction \frac{15}{-6} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{5}{2}x=\frac{6}{-6}$$

Divide 6 by -6.

$$x^{2}-\frac{5}{2}x=-1$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=-1+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-1+\frac{25}{16}$$

Add -1 to \frac{25}{16}.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{9}{16}$$

Factor x^{2}-\frac{5}{2}x+\frac{25}{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}.

$$\left(x-\frac{5}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$x-\frac{5}{4}=\frac{3}{4} x-\frac{5}{4}=-\frac{3}{4}$$

Add \frac{5}{4} to both sides of the equation.

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