Solve 75/x=x+2/3x

$$\frac { 75 } { x } = x + \frac { 2 } { 3 } x$$

Solve for x

$$x=3\sqrt{5}\approx 6.708203932$$

$$x=-3\sqrt{5}\approx -6.708203932$$

Steps by Finding Square Root

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x, the least common multiple of x,3.

$$3\times 75=3xx+\frac{2}{3}x\times 3x$$

Multiply 3 and 75 to get 225.

$$225=3xx+\frac{2}{3}x\times 3x$$

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

$$225=3x^{2}+\frac{2}{3}x\times 3x$$

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

$$225=3x^{2}+\frac{2}{3}x^{2}\times 3$$

Cancel out 3 and 3.

$$225=3x^{2}+2x^{2}$$

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$225=5x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$5x^{2}=225$$

Divide both sides by 5.

$$x^{2}=\frac{225}{5}$$

Divide 225 by 5 to get 45.

$$x^{2}=45$$

Take the square root of both sides of the equation.

$$x=3\sqrt{5} x=-3\sqrt{5}$$

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, the least common multiple of x,3.

$$3\times 75=3xx+\frac{2}{3}x\times 3x$$

Multiply 3 and 75 to get 225.

$$225=3xx+\frac{2}{3}x\times 3x$$

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

$$225=3x^{2}+\frac{2}{3}x\times 3x$$

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

$$225=3x^{2}+\frac{2}{3}x^{2}\times 3$$

Cancel out 3 and 3.

$$225=3x^{2}+2x^{2}$$

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$225=5x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$5x^{2}=225$$

Subtract 225 from both sides.

$$5x^{2}-225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 0 for b, and -225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 5\left(-225\right)}}{2\times 5}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 5\left(-225\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{0±\sqrt{-20\left(-225\right)}}{2\times 5}$$

Multiply -20 times -225.

$$x=\frac{0±\sqrt{4500}}{2\times 5}$$

Take the square root of 4500.

$$x=\frac{0±30\sqrt{5}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{0±30\sqrt{5}}{10}$$

Now solve the equation x=\frac{0±30\sqrt{5}}{10} when ± is plus.

$$x=3\sqrt{5}$$

Now solve the equation x=\frac{0±30\sqrt{5}}{10} when ± is minus.

$$x=-3\sqrt{5}$$

The equation is now solved.

$$x=3\sqrt{5} x=-3\sqrt{5}$$