Solve (9/x-2=2/x-1)

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

Solve for x

$$x=3$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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.

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

Use the distributive property to multiply x-1 by 9.

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

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

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

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

$$9x-9-2x^{2}+2x=x\times 2$$

Combine 9x and 2x to get 11x.

$$11x-9-2x^{2}=x\times 2$$

Subtract x\times 2 from both sides.

$$11x-9-2x^{2}-x\times 2=0$$

Combine 11x and -x\times 2 to get 9x.

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

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

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

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.

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

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 18.

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

Calculate the sum for each pair.

$$1+18=19 2+9=11 3+6=9$$

The solution is the pair that gives sum 9.

$$a=6 b=3$$

Rewrite -2x^{2}+9x-9 as \left(-2x^{2}+6x\right)+\left(3x-9\right).

$$\left(-2x^{2}+6x\right)+\left(3x-9\right)$$

Factor out 2x in the first and -3 in the second group.

$$2x\left(-x+3\right)-3\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(2x-3\right)$$

To find equation solutions, solve -x+3=0 and 2x-3=0.

$$x=3 x=\frac{3}{2}$$

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.

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

Use the distributive property to multiply x-1 by 9.

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

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

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

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

$$9x-9-2x^{2}+2x=x\times 2$$

Combine 9x and 2x to get 11x.

$$11x-9-2x^{2}=x\times 2$$

Subtract x\times 2 from both sides.

$$11x-9-2x^{2}-x\times 2=0$$

Combine 11x and -x\times 2 to get 9x.

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

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.

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

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

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

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-9±\sqrt{81+8\left(-9\right)}}{2\left(-2\right)}$$

Multiply 8 times -9.

$$x=\frac{-9±\sqrt{81-72}}{2\left(-2\right)}$$

Add 81 to -72.

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

Take the square root of 9.

$$x=\frac{-9±3}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-9±3}{-4}$$

Now solve the equation x=\frac{-9±3}{-4} when ± is plus. Add -9 to 3.

$$x=-\frac{6}{-4}$$

Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{-9±3}{-4} when ± is minus. Subtract 3 from -9.

$$x=-\frac{12}{-4}$$

Divide -12 by -4.

$$x=3$$

The equation is now solved.

$$x=\frac{3}{2} x=3$$

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.

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

Use the distributive property to multiply x-1 by 9.

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

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

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

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

$$9x-9-2x^{2}+2x=x\times 2$$

Combine 9x and 2x to get 11x.

$$11x-9-2x^{2}=x\times 2$$

Subtract x\times 2 from both sides.

$$11x-9-2x^{2}-x\times 2=0$$

Combine 11x and -x\times 2 to get 9x.

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

Add 9 to both sides. Anything plus zero gives itself.

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

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.

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

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide 9 by -2.

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

Divide 9 by -2.

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

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

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

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-\frac{9}{2}+\frac{81}{16}$$

Add -\frac{9}{2} to \frac{81}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{9}{16}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-\frac{9}{4}=\frac{3}{4} x-\frac{9}{4}=-\frac{3}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$x=3 x=\frac{3}{2}$$