Solve 15/x-15/1.5x=3

$$\frac{ 15 }{ x } - \frac{ 15 }{ 1.5 } x = 3$$

Solve for x

$$x = \frac{\sqrt{609} - 3}{20} \approx 1.083896268$$

$$x=\frac{-\sqrt{609}-3}{20}\approx -1.383896268$$

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 x.

$$15-\frac{2}{3}x\times 15x=3x$$

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

$$15-\frac{2}{3}x^{2}\times 15=3x$$

Express \frac{2}{3}\times 15 as a single fraction.

$$15-\frac{2\times 15}{3}x^{2}=3x$$

Multiply 2 and 15 to get 30.

$$15-\frac{30}{3}x^{2}=3x$$

Divide 30 by 3 to get 10.

$$15-10x^{2}=3x$$

Subtract 3x from both sides.

$$15-10x^{2}-3x=0$$

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.

$$-10x^{2}-3x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, -3 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-10\right)\times 15}}{2\left(-10\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-10\right)\times 15}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-\left(-3\right)±\sqrt{9+40\times 15}}{2\left(-10\right)}$$

Multiply 40 times 15.

$$x=\frac{-\left(-3\right)±\sqrt{9+600}}{2\left(-10\right)}$$

Add 9 to 600.

$$x=\frac{-\left(-3\right)±\sqrt{609}}{2\left(-10\right)}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{609}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{3±\sqrt{609}}{-20}$$

Now solve the equation x=\frac{3±\sqrt{609}}{-20} when ± is plus. Add 3 to \sqrt{609}.

$$x=\frac{\sqrt{609}+3}{-20}$$

Divide 3+\sqrt{609} by -20.

$$x=\frac{-\sqrt{609}-3}{20}$$

Now solve the equation x=\frac{3±\sqrt{609}}{-20} when ± is minus. Subtract \sqrt{609} from 3.

$$x=\frac{3-\sqrt{609}}{-20}$$

Divide 3-\sqrt{609} by -20.

$$x=\frac{\sqrt{609}-3}{20}$$

The equation is now solved.

$$x=\frac{-\sqrt{609}-3}{20} x=\frac{\sqrt{609}-3}{20}$$

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 x.

$$15-\frac{2}{3}x\times 15x=3x$$

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

$$15-\frac{2}{3}x^{2}\times 15=3x$$

Express \frac{2}{3}\times 15 as a single fraction.

$$15-\frac{2\times 15}{3}x^{2}=3x$$

Multiply 2 and 15 to get 30.

$$15-\frac{30}{3}x^{2}=3x$$

Divide 30 by 3 to get 10.

$$15-10x^{2}=3x$$

Subtract 3x from both sides.

$$15-10x^{2}-3x=0$$

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$-10x^{2}-3x=-15$$

Divide both sides by -10.

$$\frac{-10x^{2}-3x}{-10}=-\frac{15}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\left(-\frac{3}{-10}\right)x=-\frac{15}{-10}$$

Divide -3 by -10.

$$x^{2}+\frac{3}{10}x=-\frac{15}{-10}$$

Reduce the fraction \frac{-15}{-10} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{3}{10}x=\frac{3}{2}$$

Divide \frac{3}{10}, the coefficient of the x term, by 2 to get \frac{3}{20}. Then add the square of \frac{3}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{10}x+\left(\frac{3}{20}\right)^{2}=\frac{3}{2}+\left(\frac{3}{20}\right)^{2}$$

Square \frac{3}{20} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{10}x+\frac{9}{400}=\frac{3}{2}+\frac{9}{400}$$

Add \frac{3}{2} to \frac{9}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{10}x+\frac{9}{400}=\frac{609}{400}$$

Factor x^{2}+\frac{3}{10}x+\frac{9}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{20}\right)^{2}=\frac{609}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{20}\right)^{2}}=\sqrt{\frac{609}{400}}$$

Simplify.

$$x+\frac{3}{20}=\frac{\sqrt{609}}{20} x+\frac{3}{20}=-\frac{\sqrt{609}}{20}$$

Subtract \frac{3}{20} from both sides of the equation.

$$x=\frac{\sqrt{609}-3}{20} x=\frac{-\sqrt{609}-3}{20}$$