Solve 1/3x+3/15+x/5=x/15x

$$\frac { 1 } { 3 x } + \frac { 3 } { 15 } + \frac { x } { 5 } = \frac { x } { 15 x }$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{14}i}{3}\approx -0.333333333+1.247219129i$$

$$x=\frac{-\sqrt{14}i-1}{3}\approx -0.333333333-1.247219129i$$

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 15x, the least common multiple of 3x,15,5,15x.

$$5+x\times 3+3xx=x$$

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

$$5+x\times 3+3x^{2}=x$$

Subtract x from both sides.

$$5+x\times 3+3x^{2}-x=0$$

Combine x\times 3 and -x to get 2x.

$$5+2x+3x^{2}=0$$

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.

$$3x^{2}+2x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 2 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 3\times 5}}{2\times 3}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 3\times 5}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-2±\sqrt{4-12\times 5}}{2\times 3}$$

Multiply -12 times 5.

$$x=\frac{-2±\sqrt{4-60}}{2\times 3}$$

Add 4 to -60.

$$x=\frac{-2±\sqrt{-56}}{2\times 3}$$

Take the square root of -56.

$$x=\frac{-2±2\sqrt{14}i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-2±2\sqrt{14}i}{6}$$

Now solve the equation x=\frac{-2±2\sqrt{14}i}{6} when ± is plus. Add -2 to 2i\sqrt{14}.

$$x=\frac{-2+2\sqrt{14}i}{6}$$

Divide -2+2i\sqrt{14} by 6.

$$x=\frac{-1+\sqrt{14}i}{3}$$

Now solve the equation x=\frac{-2±2\sqrt{14}i}{6} when ± is minus. Subtract 2i\sqrt{14} from -2.

$$x=\frac{-2\sqrt{14}i-2}{6}$$

Divide -2-2i\sqrt{14} by 6.

$$x=\frac{-\sqrt{14}i-1}{3}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{14}i}{3} x=\frac{-\sqrt{14}i-1}{3}$$

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 15x, the least common multiple of 3x,15,5,15x.

$$5+x\times 3+3xx=x$$

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

$$5+x\times 3+3x^{2}=x$$

Subtract x from both sides.

$$5+x\times 3+3x^{2}-x=0$$

Combine x\times 3 and -x to get 2x.

$$5+2x+3x^{2}=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$2x+3x^{2}=-5$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$3x^{2}+2x=-5$$

Divide both sides by 3.

$$\frac{3x^{2}+2x}{3}=-\frac{5}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{2}{3}x=-\frac{5}{3}$$

Divide \frac{2}{3}, the coefficient of the x term, by 2 to get \frac{1}{3}. Then add the square of \frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2}{3}x+\left(\frac{1}{3}\right)^{2}=-\frac{5}{3}+\left(\frac{1}{3}\right)^{2}$$

Square \frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=-\frac{5}{3}+\frac{1}{9}$$

Add -\frac{5}{3} to \frac{1}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=-\frac{14}{9}$$

Factor x^{2}+\frac{2}{3}x+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{3}\right)^{2}=-\frac{14}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{-\frac{14}{9}}$$

Simplify.

$$x+\frac{1}{3}=\frac{\sqrt{14}i}{3} x+\frac{1}{3}=-\frac{\sqrt{14}i}{3}$$

Subtract \frac{1}{3} from both sides of the equation.

$$x=\frac{-1+\sqrt{14}i}{3} x=\frac{-\sqrt{14}i-1}{3}$$