Solve 3-27x/(x-1)(1-5x)=1

$$\frac{ 3-27x }{ (x-1)(1-5x) } =1$$

Solve for x

$$x = \frac{\sqrt{1009} + 33}{10} \approx 6.476476035$$

$$x=\frac{33-\sqrt{1009}}{10}\approx 0.123523965$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{1}{5},1 since division by zero is not defined. Multiply both sides of the equation by \left(5x-1\right)\left(-x+1\right).

$$3-27x=\left(5x-1\right)\left(-x+1\right)$$

Use the distributive property to multiply 5x-1 by -x+1 and combine like terms.

$$3-27x=-5x^{2}+6x-1$$

Add 5x^{2} to both sides.

$$3-27x+5x^{2}=6x-1$$

Subtract 6x from both sides.

$$3-27x+5x^{2}-6x=-1$$

Combine -27x and -6x to get -33x.

$$3-33x+5x^{2}=-1$$

Add 1 to both sides.

$$3-33x+5x^{2}+1=0$$

Add 3 and 1 to get 4.

$$4-33x+5x^{2}=0$$

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.

$$5x^{2}-33x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -33 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 5\times 4}}{2\times 5}$$

Square -33.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\times 5\times 4}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-33\right)±\sqrt{1089-20\times 4}}{2\times 5}$$

Multiply -20 times 4.

$$x=\frac{-\left(-33\right)±\sqrt{1089-80}}{2\times 5}$$

Add 1089 to -80.

$$x=\frac{-\left(-33\right)±\sqrt{1009}}{2\times 5}$$

The opposite of -33 is 33.

$$x=\frac{33±\sqrt{1009}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{33±\sqrt{1009}}{10}$$

Now solve the equation x=\frac{33±\sqrt{1009}}{10} when ± is plus. Add 33 to \sqrt{1009}.

$$x=\frac{\sqrt{1009}+33}{10}$$

Now solve the equation x=\frac{33±\sqrt{1009}}{10} when ± is minus. Subtract \sqrt{1009} from 33.

$$x=\frac{33-\sqrt{1009}}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{1009}+33}{10} x=\frac{33-\sqrt{1009}}{10}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{1}{5},1 since division by zero is not defined. Multiply both sides of the equation by \left(5x-1\right)\left(-x+1\right).

$$3-27x=\left(5x-1\right)\left(-x+1\right)$$

Use the distributive property to multiply 5x-1 by -x+1 and combine like terms.

$$3-27x=-5x^{2}+6x-1$$

Add 5x^{2} to both sides.

$$3-27x+5x^{2}=6x-1$$

Subtract 6x from both sides.

$$3-27x+5x^{2}-6x=-1$$

Combine -27x and -6x to get -33x.

$$3-33x+5x^{2}=-1$$

Subtract 3 from both sides.

$$-33x+5x^{2}=-1-3$$

Subtract 3 from -1 to get -4.

$$-33x+5x^{2}=-4$$

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.

$$5x^{2}-33x=-4$$

Divide both sides by 5.

$$\frac{5x^{2}-33x}{5}=-\frac{4}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{33}{5}x=-\frac{4}{5}$$

Divide -\frac{33}{5}, the coefficient of the x term, by 2 to get -\frac{33}{10}. Then add the square of -\frac{33}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{33}{5}x+\left(-\frac{33}{10}\right)^{2}=-\frac{4}{5}+\left(-\frac{33}{10}\right)^{2}$$

Square -\frac{33}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{33}{5}x+\frac{1089}{100}=-\frac{4}{5}+\frac{1089}{100}$$

Add -\frac{4}{5} to \frac{1089}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{33}{5}x+\frac{1089}{100}=\frac{1009}{100}$$

Factor x^{2}-\frac{33}{5}x+\frac{1089}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{33}{10}\right)^{2}=\frac{1009}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{33}{10}\right)^{2}}=\sqrt{\frac{1009}{100}}$$

Simplify.

$$x-\frac{33}{10}=\frac{\sqrt{1009}}{10} x-\frac{33}{10}=-\frac{\sqrt{1009}}{10}$$

Add \frac{33}{10} to both sides of the equation.

$$x=\frac{\sqrt{1009}+33}{10} x=\frac{33-\sqrt{1009}}{10}$$