Solve 24*x^2+1/x^2=27

$$24 \cdot x ^ { 2 } + \frac { 1 } { x ^ { 2 } } = 27$$

Solve for x

$$x = -\frac{\sqrt{3 \sqrt{633} + 81}}{12} \approx -1.042427968$$

$$x = \frac{\sqrt{3 \sqrt{633} + 81}}{12} \approx 1.042427968$$

$$x=\frac{\sqrt{81-3\sqrt{633}}}{12}\approx 0.195816067$$

$$x=-\frac{\sqrt{81-3\sqrt{633}}}{12}\approx -0.195816067$$

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 x^{2}.

$$24x^{2}x^{2}+1=27x^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$24x^{4}+1=27x^{2}$$

Subtract 27x^{2} from both sides.

$$24x^{4}+1-27x^{2}=0$$

Substitute t for x^{2}.

$$24t^{2}-27t+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 24 for a, -27 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 24\times 1}}{2\times 24}$$

Do the calculations.

$$t=\frac{27±\sqrt{633}}{48}$$

Solve the equation t=\frac{27±\sqrt{633}}{48} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{633}}{48}+\frac{9}{16} t=-\frac{\sqrt{633}}{48}+\frac{9}{16}$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=\frac{\sqrt{\frac{\sqrt{633}}{3}+9}}{4} x=-\frac{\sqrt{\frac{\sqrt{633}}{3}+9}}{4} x=\frac{\sqrt{-\frac{\sqrt{633}}{3}+9}}{4} x=-\frac{\sqrt{-\frac{\sqrt{633}}{3}+9}}{4}$$