Solve 2x-1/3x=4

$$2x- \frac{ 1 }{ 3x } =4$$

Solve for x

$$x=\frac{\sqrt{42}}{6}+1\approx 2.08012345$$

$$x=-\frac{\sqrt{42}}{6}+1\approx -0.08012345$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x.

$$2x\times 3x-1=12x$$

Multiply 2 and 3 to get 6.

$$6xx-1=12x$$

Multiply x and x to get x^{2}.

$$6x^{2}-1=12x$$

Subtract 12x from both sides.

$$6x^{2}-1-12x=0$$

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.

$$6x^{2}-12x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -12 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 6\left(-1\right)}}{2\times 6}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 6\left(-1\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-12\right)±\sqrt{144-24\left(-1\right)}}{2\times 6}$$

Multiply -24 times -1.

$$x=\frac{-\left(-12\right)±\sqrt{144+24}}{2\times 6}$$

Add 144 to 24.

$$x=\frac{-\left(-12\right)±\sqrt{168}}{2\times 6}$$

Take the square root of 168.

$$x=\frac{-\left(-12\right)±2\sqrt{42}}{2\times 6}$$

The opposite of -12 is 12.

$$x=\frac{12±2\sqrt{42}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{12±2\sqrt{42}}{12}$$

Now solve the equation x=\frac{12±2\sqrt{42}}{12} when ± is plus. Add 12 to 2\sqrt{42}.

$$x=\frac{2\sqrt{42}+12}{12}$$

Divide 12+2\sqrt{42} by 12.

$$x=\frac{\sqrt{42}}{6}+1$$

Now solve the equation x=\frac{12±2\sqrt{42}}{12} when ± is minus. Subtract 2\sqrt{42} from 12.

$$x=\frac{12-2\sqrt{42}}{12}$$

Divide 12-2\sqrt{42} by 12.

$$x=-\frac{\sqrt{42}}{6}+1$$

The equation is now solved.

$$x=\frac{\sqrt{42}}{6}+1 x=-\frac{\sqrt{42}}{6}+1$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x.

$$2x\times 3x-1=12x$$

Multiply 2 and 3 to get 6.

$$6xx-1=12x$$

Multiply x and x to get x^{2}.

$$6x^{2}-1=12x$$

Subtract 12x from both sides.

$$6x^{2}-1-12x=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$6x^{2}-12x=1$$

Divide both sides by 6.

$$\frac{6x^{2}-12x}{6}=\frac{1}{6}$$

Dividing by 6 undoes the multiplication by 6.

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

Divide -12 by 6.

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

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

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

Add \frac{1}{6} to 1.

$$x^{2}-2x+1=\frac{7}{6}$$

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}.

$$\left(x-1\right)^{2}=\frac{7}{6}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{7}{6}}$$

Simplify.

$$x-1=\frac{\sqrt{42}}{6} x-1=-\frac{\sqrt{42}}{6}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{42}}{6}+1 x=-\frac{\sqrt{42}}{6}+1$$