Solve 2left(3x-1right)3left(2x-1right)=4

$$2 \left( 3x-1 \right) 3 \left( 2x-1 \right) =4$$

Solve for x

$$x=\frac{\sqrt{17}+5}{12}\approx 0.760258802$$

$$x=\frac{5-\sqrt{17}}{12}\approx 0.073074531$$

Steps Using the Quadratic Formula

Multiply 2 and 3 to get 6.

$$6\left(3x-1\right)\left(2x-1\right)=4$$

Use the distributive property to multiply 6 by 3x-1.

$$\left(18x-6\right)\left(2x-1\right)=4$$

Use the distributive property to multiply 18x-6 by 2x-1 and combine like terms.

$$36x^{2}-30x+6=4$$

Subtract 4 from both sides.

$$36x^{2}-30x+6-4=0$$

Subtract 4 from 6 to get 2.

$$36x^{2}-30x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, -30 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 36\times 2}}{2\times 36}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 36\times 2}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-\left(-30\right)±\sqrt{900-144\times 2}}{2\times 36}$$

Multiply -144 times 2.

$$x=\frac{-\left(-30\right)±\sqrt{900-288}}{2\times 36}$$

Add 900 to -288.

$$x=\frac{-\left(-30\right)±\sqrt{612}}{2\times 36}$$

Take the square root of 612.

$$x=\frac{-\left(-30\right)±6\sqrt{17}}{2\times 36}$$

The opposite of -30 is 30.

$$x=\frac{30±6\sqrt{17}}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{30±6\sqrt{17}}{72}$$

Now solve the equation x=\frac{30±6\sqrt{17}}{72} when ± is plus. Add 30 to 6\sqrt{17}.

$$x=\frac{6\sqrt{17}+30}{72}$$

Divide 30+6\sqrt{17} by 72.

$$x=\frac{\sqrt{17}+5}{12}$$

Now solve the equation x=\frac{30±6\sqrt{17}}{72} when ± is minus. Subtract 6\sqrt{17} from 30.

$$x=\frac{30-6\sqrt{17}}{72}$$

Divide 30-6\sqrt{17} by 72.

$$x=\frac{5-\sqrt{17}}{12}$$

The equation is now solved.

$$x=\frac{\sqrt{17}+5}{12} x=\frac{5-\sqrt{17}}{12}$$

Steps for Completing the Square

Multiply 2 and 3 to get 6.

$$6\left(3x-1\right)\left(2x-1\right)=4$$

Use the distributive property to multiply 6 by 3x-1.

$$\left(18x-6\right)\left(2x-1\right)=4$$

Use the distributive property to multiply 18x-6 by 2x-1 and combine like terms.

$$36x^{2}-30x+6=4$$

Subtract 6 from both sides.

$$36x^{2}-30x=4-6$$

Subtract 6 from 4 to get -2.

$$36x^{2}-30x=-2$$

Divide both sides by 36.

$$\frac{36x^{2}-30x}{36}=-\frac{2}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\left(-\frac{30}{36}\right)x=-\frac{2}{36}$$

Reduce the fraction \frac{-30}{36} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{5}{6}x=-\frac{2}{36}$$

Reduce the fraction \frac{-2}{36} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{6}x=-\frac{1}{18}$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=-\frac{1}{18}+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=-\frac{1}{18}+\frac{25}{144}$$

Add -\frac{1}{18} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{17}{144}$$

Factor x^{2}-\frac{5}{6}x+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{12}\right)^{2}=\frac{17}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{\frac{17}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{\sqrt{17}}{12} x-\frac{5}{12}=-\frac{\sqrt{17}}{12}$$

Add \frac{5}{12} to both sides of the equation.

$$x=\frac{\sqrt{17}+5}{12} x=\frac{5-\sqrt{17}}{12}$$