Solve 3x-1/5x=1/10

$$3 x - \frac { 1 } { 5 x } = \frac { 1 } { 10 }$$

Solve for x

$$x=\frac{\sqrt{241}+1}{60}\approx 0.275402912$$

$$x=\frac{1-\sqrt{241}}{60}\approx -0.242069578$$

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 10x, the least common multiple of 5x,10.

$$3x\times 10x-2=x$$

Multiply 3 and 10 to get 30.

$$30xx-2=x$$

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

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

Subtract x from both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 30 for a, -1 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 30\left(-2\right)}}{2\times 30}$$

Multiply -4 times 30.

$$x=\frac{-\left(-1\right)±\sqrt{1-120\left(-2\right)}}{2\times 30}$$

Multiply -120 times -2.

$$x=\frac{-\left(-1\right)±\sqrt{1+240}}{2\times 30}$$

Add 1 to 240.

$$x=\frac{-\left(-1\right)±\sqrt{241}}{2\times 30}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{241}}{2\times 30}$$

Multiply 2 times 30.

$$x=\frac{1±\sqrt{241}}{60}$$

Now solve the equation x=\frac{1±\sqrt{241}}{60} when ± is plus. Add 1 to \sqrt{241}.

$$x=\frac{\sqrt{241}+1}{60}$$

Now solve the equation x=\frac{1±\sqrt{241}}{60} when ± is minus. Subtract \sqrt{241} from 1.

$$x=\frac{1-\sqrt{241}}{60}$$

The equation is now solved.

$$x=\frac{\sqrt{241}+1}{60} x=\frac{1-\sqrt{241}}{60}$$

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 10x, the least common multiple of 5x,10.

$$3x\times 10x-2=x$$

Multiply 3 and 10 to get 30.

$$30xx-2=x$$

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

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

Subtract x from both sides.

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

Add 2 to both sides. Anything plus zero gives itself.

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

Divide both sides by 30.

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

Dividing by 30 undoes the multiplication by 30.

$$x^{2}-\frac{1}{30}x=\frac{2}{30}$$

Reduce the fraction \frac{2}{30} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{30}x=\frac{1}{15}$$

Divide -\frac{1}{30}, the coefficient of the x term, by 2 to get -\frac{1}{60}. Then add the square of -\frac{1}{60} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{30}x+\left(-\frac{1}{60}\right)^{2}=\frac{1}{15}+\left(-\frac{1}{60}\right)^{2}$$

Square -\frac{1}{60} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{30}x+\frac{1}{3600}=\frac{1}{15}+\frac{1}{3600}$$

Add \frac{1}{15} to \frac{1}{3600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{30}x+\frac{1}{3600}=\frac{241}{3600}$$

Factor x^{2}-\frac{1}{30}x+\frac{1}{3600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{60}\right)^{2}=\frac{241}{3600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{60}\right)^{2}}=\sqrt{\frac{241}{3600}}$$

Simplify.

$$x-\frac{1}{60}=\frac{\sqrt{241}}{60} x-\frac{1}{60}=-\frac{\sqrt{241}}{60}$$

Add \frac{1}{60} to both sides of the equation.

$$x=\frac{\sqrt{241}+1}{60} x=\frac{1-\sqrt{241}}{60}$$