Solve x-1/27=3/x-1

$$\frac { x - 1 } { 27 } = \frac { 3 } { x - 1 }$$

Solve for x

$$x=-8$$

$$x=10$$

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 27\left(x-1\right), the least common multiple of 27,x-1.

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

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

$$\left(x-1\right)^{2}=27\times 3$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1=27\times 3$$

Multiply 27 and 3 to get 81.

$$x^{2}-2x+1=81$$

Subtract 81 from both sides.

$$x^{2}-2x+1-81=0$$

Subtract 81 from 1 to get -80.

$$x^{2}-2x-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-80\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-80\right)}}{2}$$

Multiply -4 times -80.

$$x=\frac{-\left(-2\right)±\sqrt{4+320}}{2}$$

Add 4 to 320.

$$x=\frac{-\left(-2\right)±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-\left(-2\right)±18}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±18}{2}$$

Now solve the equation x=\frac{2±18}{2} when ± is plus. Add 2 to 18.

$$x=\frac{20}{2}$$

Divide 20 by 2.

$$x=10$$

Now solve the equation x=\frac{2±18}{2} when ± is minus. Subtract 18 from 2.

$$x=-\frac{16}{2}$$

Divide -16 by 2.

$$x=-8$$

The equation is now solved.

$$x=10 x=-8$$

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 27\left(x-1\right), the least common multiple of 27,x-1.

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

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

$$\left(x-1\right)^{2}=27\times 3$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1=27\times 3$$

Multiply 27 and 3 to get 81.

$$x^{2}-2x+1=81$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{81}$$

Simplify.

$$x-1=9 x-1=-9$$

Add 1 to both sides of the equation.

$$x=10 x=-8$$