Solve 3x=27/9x-1

$$3 x = \frac { 27 } { 9 x - 1 }$$

Solve for x

$$x = \frac{5 \sqrt{13} + 1}{18} \approx 1.057097577$$

$$x=\frac{1-5\sqrt{13}}{18}\approx -0.945986465$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{1}{9} since division by zero is not defined. Multiply both sides of the equation by 9x-1.

$$3x\left(9x-1\right)=27$$

Use the distributive property to multiply 3x by 9x-1.

$$27x^{2}-3x=27$$

Subtract 27 from both sides.

$$27x^{2}-3x-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 27 for a, -3 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 27\left(-27\right)}}{2\times 27}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 27\left(-27\right)}}{2\times 27}$$

Multiply -4 times 27.

$$x=\frac{-\left(-3\right)±\sqrt{9-108\left(-27\right)}}{2\times 27}$$

Multiply -108 times -27.

$$x=\frac{-\left(-3\right)±\sqrt{9+2916}}{2\times 27}$$

Add 9 to 2916.

$$x=\frac{-\left(-3\right)±\sqrt{2925}}{2\times 27}$$

Take the square root of 2925.

$$x=\frac{-\left(-3\right)±15\sqrt{13}}{2\times 27}$$

The opposite of -3 is 3.

$$x=\frac{3±15\sqrt{13}}{2\times 27}$$

Multiply 2 times 27.

$$x=\frac{3±15\sqrt{13}}{54}$$

Now solve the equation x=\frac{3±15\sqrt{13}}{54} when ± is plus. Add 3 to 15\sqrt{13}.

$$x=\frac{15\sqrt{13}+3}{54}$$

Divide 3+15\sqrt{13} by 54.

$$x=\frac{5\sqrt{13}+1}{18}$$

Now solve the equation x=\frac{3±15\sqrt{13}}{54} when ± is minus. Subtract 15\sqrt{13} from 3.

$$x=\frac{3-15\sqrt{13}}{54}$$

Divide 3-15\sqrt{13} by 54.

$$x=\frac{1-5\sqrt{13}}{18}$$

The equation is now solved.

$$x=\frac{5\sqrt{13}+1}{18} x=\frac{1-5\sqrt{13}}{18}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{1}{9} since division by zero is not defined. Multiply both sides of the equation by 9x-1.

$$3x\left(9x-1\right)=27$$

Use the distributive property to multiply 3x by 9x-1.

$$27x^{2}-3x=27$$

Divide both sides by 27.

$$\frac{27x^{2}-3x}{27}=\frac{27}{27}$$

Dividing by 27 undoes the multiplication by 27.

$$x^{2}+\left(-\frac{3}{27}\right)x=\frac{27}{27}$$

Reduce the fraction \frac{-3}{27} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{1}{9}x=\frac{27}{27}$$

Divide 27 by 27.

$$x^{2}-\frac{1}{9}x=1$$

Divide -\frac{1}{9}, the coefficient of the x term, by 2 to get -\frac{1}{18}. Then add the square of -\frac{1}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{9}x+\left(-\frac{1}{18}\right)^{2}=1+\left(-\frac{1}{18}\right)^{2}$$

Square -\frac{1}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{9}x+\frac{1}{324}=1+\frac{1}{324}$$

Add 1 to \frac{1}{324}.

$$x^{2}-\frac{1}{9}x+\frac{1}{324}=\frac{325}{324}$$

Factor x^{2}-\frac{1}{9}x+\frac{1}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{18}\right)^{2}=\frac{325}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{18}\right)^{2}}=\sqrt{\frac{325}{324}}$$

Simplify.

$$x-\frac{1}{18}=\frac{5\sqrt{13}}{18} x-\frac{1}{18}=-\frac{5\sqrt{13}}{18}$$

Add \frac{1}{18} to both sides of the equation.

$$x=\frac{5\sqrt{13}+1}{18} x=\frac{1-5\sqrt{13}}{18}$$