Solve 1/-x^2+14=26

$$\frac { 1 } { - x ^ { 2 } } + 14 = 26$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{3}i}{6}\approx -0-0.288675135i$$

$$x=\frac{\sqrt{3}i}{6}\approx 0.288675135i$$

Steps by Finding Square Root

Factor -x^{2}.

$$\frac{1}{-x^{2}}+14=26$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 14 times \frac{-x^{2}}{-x^{2}}.

$$\frac{1}{-x^{2}}+\frac{14\left(-1\right)x^{2}}{-x^{2}}=26$$

Since \frac{1}{-x^{2}} and \frac{14\left(-1\right)x^{2}}{-x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{1+14\left(-1\right)x^{2}}{-x^{2}}=26$$

Do the multiplications in 1+14\left(-1\right)x^{2}.

$$\frac{1-14x^{2}}{-x^{2}}=26$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}.

$$-\left(1-14x^{2}\right)=26x^{2}$$

To find the opposite of 1-14x^{2}, find the opposite of each term.

$$-1+14x^{2}=26x^{2}$$

Subtract 26x^{2} from both sides.

$$-1+14x^{2}-26x^{2}=0$$

Combine 14x^{2} and -26x^{2} to get -12x^{2}.

$$-1-12x^{2}=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$-12x^{2}=1$$

Divide both sides by -12.

$$x^{2}=-\frac{1}{12}$$

The equation is now solved.

$$x=\frac{\sqrt{3}i}{6} x=-\frac{\sqrt{3}i}{6}$$

Steps Using the Quadratic Formula

Factor -x^{2}.

$$\frac{1}{-x^{2}}+14=26$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 14 times \frac{-x^{2}}{-x^{2}}.

$$\frac{1}{-x^{2}}+\frac{14\left(-1\right)x^{2}}{-x^{2}}=26$$

Since \frac{1}{-x^{2}} and \frac{14\left(-1\right)x^{2}}{-x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{1+14\left(-1\right)x^{2}}{-x^{2}}=26$$

Do the multiplications in 1+14\left(-1\right)x^{2}.

$$\frac{1-14x^{2}}{-x^{2}}=26$$

Subtract 26 from both sides.

$$\frac{1-14x^{2}}{-x^{2}}-26=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 26 times \frac{-x^{2}}{-x^{2}}.

$$\frac{1-14x^{2}}{-x^{2}}-\frac{26\left(-1\right)x^{2}}{-x^{2}}=0$$

Since \frac{1-14x^{2}}{-x^{2}} and \frac{26\left(-1\right)x^{2}}{-x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1-14x^{2}-26\left(-1\right)x^{2}}{-x^{2}}=0$$

Do the multiplications in 1-14x^{2}-26\left(-1\right)x^{2}.

$$\frac{1-14x^{2}+26x^{2}}{-x^{2}}=0$$

Combine like terms in 1-14x^{2}+26x^{2}.

$$\frac{1+12x^{2}}{-x^{2}}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}.

$$-\left(1+12x^{2}\right)=0$$

Divide both sides by -1. Zero divided by any non-zero number gives zero.

$$1+12x^{2}=0$$

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.

$$12x^{2}+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 0 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 12}}{2\times 12}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 12}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{0±\sqrt{-48}}{2\times 12}$$

Take the square root of -48.

$$x=\frac{0±4\sqrt{3}i}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{0±4\sqrt{3}i}{24}$$

Now solve the equation x=\frac{0±4\sqrt{3}i}{24} when ± is plus.

$$x=\frac{\sqrt{3}i}{6}$$

Now solve the equation x=\frac{0±4\sqrt{3}i}{24} when ± is minus.

$$x=-\frac{\sqrt{3}i}{6}$$

The equation is now solved.

$$x=\frac{\sqrt{3}i}{6} x=-\frac{\sqrt{3}i}{6}$$