Solve 2x+1/3x=5

$$2 x + \frac { 1 } { 3 x } = 5$$

Solve for x

$$x=\frac{\sqrt{201}}{12}+\frac{5}{4}\approx 2.431453907$$

$$x=-\frac{\sqrt{201}}{12}+\frac{5}{4}\approx 0.068546093$$

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.

$$2x\times 3x+1=15x$$

Multiply 2 and 3 to get 6.

$$6xx+1=15x$$

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

$$6x^{2}+1=15x$$

Subtract 15x from both sides.

$$6x^{2}+1-15x=0$$

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.

$$6x^{2}-15x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -15 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 6}}{2\times 6}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 6}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-15\right)±\sqrt{225-24}}{2\times 6}$$

Add 225 to -24.

$$x=\frac{-\left(-15\right)±\sqrt{201}}{2\times 6}$$

The opposite of -15 is 15.

$$x=\frac{15±\sqrt{201}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{15±\sqrt{201}}{12}$$

Now solve the equation x=\frac{15±\sqrt{201}}{12} when ± is plus. Add 15 to \sqrt{201}.

$$x=\frac{\sqrt{201}+15}{12}$$

Divide 15+\sqrt{201} by 12.

$$x=\frac{\sqrt{201}}{12}+\frac{5}{4}$$

Now solve the equation x=\frac{15±\sqrt{201}}{12} when ± is minus. Subtract \sqrt{201} from 15.

$$x=\frac{15-\sqrt{201}}{12}$$

Divide 15-\sqrt{201} by 12.

$$x=-\frac{\sqrt{201}}{12}+\frac{5}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{201}}{12}+\frac{5}{4} x=-\frac{\sqrt{201}}{12}+\frac{5}{4}$$

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.

$$2x\times 3x+1=15x$$

Multiply 2 and 3 to get 6.

$$6xx+1=15x$$

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

$$6x^{2}+1=15x$$

Subtract 15x from both sides.

$$6x^{2}+1-15x=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$6x^{2}-15x=-1$$

Divide both sides by 6.

$$\frac{6x^{2}-15x}{6}=-\frac{1}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{15}{6}\right)x=-\frac{1}{6}$$

Reduce the fraction \frac{-15}{6} to lowest terms by extracting and canceling out 3.

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

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=-\frac{1}{6}+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-\frac{1}{6}+\frac{25}{16}$$

Add -\frac{1}{6} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{67}{48}$$

Factor x^{2}-\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{4}\right)^{2}=\frac{67}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{67}{48}}$$

Simplify.

$$x-\frac{5}{4}=\frac{\sqrt{201}}{12} x-\frac{5}{4}=-\frac{\sqrt{201}}{12}$$

Add \frac{5}{4} to both sides of the equation.

$$x=\frac{\sqrt{201}}{12}+\frac{5}{4} x=-\frac{\sqrt{201}}{12}+\frac{5}{4}$$