Solve 9-x/x-1=4x

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

Solve for x

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

Steps Using Factoring By Grouping

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

$$\left(x-1\right)\times 9-x=4x\left(x-1\right)$$

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

$$9x-9-x=4x\left(x-1\right)$$

Use the distributive property to multiply 4x by x-1.

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

Subtract 4x^{2} from both sides.

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

Add 4x to both sides.

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

Combine 9x and 4x to get 13x.

$$13x-9-x-4x^{2}=0$$

Combine 13x and -x to get 12x.

$$12x-9-4x^{2}=0$$

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

$$-4x^{2}+12x-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=12 ab=-4\left(-9\right)=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 12.

$$a=6 b=6$$

Rewrite -4x^{2}+12x-9 as \left(-4x^{2}+6x\right)+\left(6x-9\right).

$$\left(-4x^{2}+6x\right)+\left(6x-9\right)$$

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

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

Factor out common term 2x-3 by using distributive property.

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

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

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

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

$$\left(x-1\right)\times 9-x=4x\left(x-1\right)$$

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

$$9x-9-x=4x\left(x-1\right)$$

Use the distributive property to multiply 4x by x-1.

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

Subtract 4x^{2} from both sides.

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

Add 4x to both sides.

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

Combine 9x and 4x to get 13x.

$$13x-9-x-4x^{2}=0$$

Combine 13x and -x to get 12x.

$$12x-9-4x^{2}=0$$

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.

$$-4x^{2}+12x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 12 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-4\right)\left(-9\right)}}{2\left(-4\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-4\right)\left(-9\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-12±\sqrt{144+16\left(-9\right)}}{2\left(-4\right)}$$

Multiply 16 times -9.

$$x=\frac{-12±\sqrt{144-144}}{2\left(-4\right)}$$

Add 144 to -144.

$$x=\frac{-12±\sqrt{0}}{2\left(-4\right)}$$

Take the square root of 0.

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

Multiply 2 times -4.

$$x=-\frac{12}{-8}$$

Reduce the fraction \frac{-12}{-8} to lowest terms by extracting and canceling out 4.

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

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

$$\left(x-1\right)\times 9-x=4x\left(x-1\right)$$

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

$$9x-9-x=4x\left(x-1\right)$$

Use the distributive property to multiply 4x by x-1.

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

Subtract 4x^{2} from both sides.

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

Add 4x to both sides.

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

Combine 9x and 4x to get 13x.

$$13x-9-x-4x^{2}=0$$

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

$$13x-x-4x^{2}=9$$

Combine 13x and -x to get 12x.

$$12x-4x^{2}=9$$

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.

$$-4x^{2}+12x=9$$

Divide both sides by -4.

$$\frac{-4x^{2}+12x}{-4}=\frac{9}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{12}{-4}x=\frac{9}{-4}$$

Divide 12 by -4.

$$x^{2}-3x=\frac{9}{-4}$$

Divide 9 by -4.

$$x^{2}-3x=-\frac{9}{4}$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=\frac{-9+9}{4}$$

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

$$x^{2}-3x+\frac{9}{4}=0$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-\frac{3}{2}=0 x-\frac{3}{2}=0$$

Add \frac{3}{2} to both sides of the equation.

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

The equation is now solved. Solutions are the same.

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