Solve x=90/x+90/2x

$$x= \frac{ 90 }{ x } + \frac{ 90 }{ 2x }$$

Solve for x

$$x=3\sqrt{15}\approx 11.618950039$$

$$x=-3\sqrt{15}\approx -11.618950039$$

Steps by Finding Square Root

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 2x is 2x. Multiply \frac{90}{x} times \frac{2}{2}.

$$x=\frac{90\times 2}{2x}+\frac{90}{2x}$$

Since \frac{90\times 2}{2x} and \frac{90}{2x} have the same denominator, add them by adding their numerators.

$$x=\frac{90\times 2+90}{2x}$$

Do the multiplications in 90\times 2+90.

$$x=\frac{180+90}{2x}$$

Do the calculations in 180+90.

$$x=\frac{270}{2x}$$

Subtract \frac{270}{2x} from both sides.

$$x-\frac{270}{2x}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2x}{2x}.

$$\frac{x\times 2x}{2x}-\frac{270}{2x}=0$$

Since \frac{x\times 2x}{2x} and \frac{270}{2x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times 2x-270}{2x}=0$$

Do the multiplications in x\times 2x-270.

$$\frac{2x^{2}-270}{2x}=0$$

Factor the expressions that are not already factored in \frac{2x^{2}-270}{2x}.

$$\frac{2\left(x^{2}-135\right)}{2x}=0$$

Cancel out 2 in both numerator and denominator.

$$\frac{x^{2}-135}{x}=0$$

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

$$x^{2}-135=0$$

Add 135 to both sides. Anything plus zero gives itself.

$$x^{2}=135$$

Take the square root of both sides of the equation.

$$x=3\sqrt{15} x=-3\sqrt{15}$$

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 2x is 2x. Multiply \frac{90}{x} times \frac{2}{2}.

$$x=\frac{90\times 2}{2x}+\frac{90}{2x}$$

Since \frac{90\times 2}{2x} and \frac{90}{2x} have the same denominator, add them by adding their numerators.

$$x=\frac{90\times 2+90}{2x}$$

Do the multiplications in 90\times 2+90.

$$x=\frac{180+90}{2x}$$

Do the calculations in 180+90.

$$x=\frac{270}{2x}$$

Subtract \frac{270}{2x} from both sides.

$$x-\frac{270}{2x}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2x}{2x}.

$$\frac{x\times 2x}{2x}-\frac{270}{2x}=0$$

Since \frac{x\times 2x}{2x} and \frac{270}{2x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times 2x-270}{2x}=0$$

Do the multiplications in x\times 2x-270.

$$\frac{2x^{2}-270}{2x}=0$$

Factor the expressions that are not already factored in \frac{2x^{2}-270}{2x}.

$$\frac{2\left(x^{2}-135\right)}{2x}=0$$

Cancel out 2 in both numerator and denominator.

$$\frac{x^{2}-135}{x}=0$$

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

$$x^{2}-135=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -135 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-135\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-135\right)}}{2}$$

Multiply -4 times -135.

$$x=\frac{0±\sqrt{540}}{2}$$

Take the square root of 540.

$$x=\frac{0±6\sqrt{15}}{2}$$

Now solve the equation x=\frac{0±6\sqrt{15}}{2} when ± is plus.

$$x=3\sqrt{15}$$

Now solve the equation x=\frac{0±6\sqrt{15}}{2} when ± is minus.

$$x=-3\sqrt{15}$$

The equation is now solved.

$$x=3\sqrt{15} x=-3\sqrt{15}$$