Solve 1/2+3/x^2=14

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

Solve for x

$$x=\frac{\sqrt{2}}{3}\approx 0.471404521$$

$$x=-\frac{\sqrt{2}}{3}\approx -0.471404521$$

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^{2}, the least common multiple of 2,x^{2}.

$$2x^{2}\times \frac{1}{2}+2\times 3=28x^{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$x^{2}+2\times 3=28x^{2}$$

Multiply 2 and 3 to get 6.

$$x^{2}+6=28x^{2}$$

Subtract 28x^{2} from both sides.

$$x^{2}+6-28x^{2}=0$$

Combine x^{2} and -28x^{2} to get -27x^{2}.

$$-27x^{2}+6=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$-27x^{2}=-6$$

Divide both sides by -27.

$$x^{2}=\frac{-6}{-27}$$

Reduce the fraction \frac{-6}{-27} to lowest terms by extracting and canceling out -3.

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

Take the square root of both sides of the equation.

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

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^{2}, the least common multiple of 2,x^{2}.

$$2x^{2}\times \frac{1}{2}+2\times 3=28x^{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$x^{2}+2\times 3=28x^{2}$$

Multiply 2 and 3 to get 6.

$$x^{2}+6=28x^{2}$$

Subtract 28x^{2} from both sides.

$$x^{2}+6-28x^{2}=0$$

Combine x^{2} and -28x^{2} to get -27x^{2}.

$$-27x^{2}+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -27 for a, 0 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-27\right)\times 6}}{2\left(-27\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-27\right)\times 6}}{2\left(-27\right)}$$

Multiply -4 times -27.

$$x=\frac{0±\sqrt{108\times 6}}{2\left(-27\right)}$$

Multiply 108 times 6.

$$x=\frac{0±\sqrt{648}}{2\left(-27\right)}$$

Take the square root of 648.

$$x=\frac{0±18\sqrt{2}}{2\left(-27\right)}$$

Multiply 2 times -27.

$$x=\frac{0±18\sqrt{2}}{-54}$$

Now solve the equation x=\frac{0±18\sqrt{2}}{-54} when ± is plus.

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

Now solve the equation x=\frac{0±18\sqrt{2}}{-54} when ± is minus.

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

The equation is now solved.

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