Solve 6/x-3/2x=x

$$\frac { 6 } { x } - \frac { 3 } { 2 x } = x$$

Solve for x

$$x = \frac{3 \sqrt{2}}{2} \approx 2.121320344$$

$$x = -\frac{3 \sqrt{2}}{2} \approx -2.121320344$$

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 2x, the least common multiple of x,2x.

$$2\times 6-3=x\times 2x$$

Multiply 2 and 6 to get 12.

$$12-3=x\times 2x$$

Subtract 3 from 12 to get 9.

$$9=x\times 2x$$

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

$$9=x^{2}\times 2$$

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

$$x^{2}\times 2=9$$

Divide both sides by 2.

$$x^{2}=\frac{9}{2}$$

Take the square root of both sides of the equation.

$$x=\frac{3\sqrt{2}}{2} x=-\frac{3\sqrt{2}}{2}$$

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 2x, the least common multiple of x,2x.

$$2\times 6-3=x\times 2x$$

Multiply 2 and 6 to get 12.

$$12-3=x\times 2x$$

Subtract 3 from 12 to get 9.

$$9=x\times 2x$$

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

$$9=x^{2}\times 2$$

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

$$x^{2}\times 2=9$$

Subtract 9 from both sides.

$$x^{2}\times 2-9=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$2x^{2}-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 0 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 2\left(-9\right)}}{2\times 2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 2\left(-9\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{0±\sqrt{-8\left(-9\right)}}{2\times 2}$$

Multiply -8 times -9.

$$x=\frac{0±\sqrt{72}}{2\times 2}$$

Take the square root of 72.

$$x=\frac{0±6\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{0±6\sqrt{2}}{4}$$

Now solve the equation x=\frac{0±6\sqrt{2}}{4} when ± is plus.

$$x=\frac{3\sqrt{2}}{2}$$

Now solve the equation x=\frac{0±6\sqrt{2}}{4} when ± is minus.

$$x=-\frac{3\sqrt{2}}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{2}}{2} x=-\frac{3\sqrt{2}}{2}$$