Solve 15x^4-114x^2+4=0

$$15 x ^ { 4 } - 114 x ^ { 2 } + 4 = 0$$

Solve for x

$$x=-\frac{\sqrt{855-15\sqrt{3189}}}{15}\approx -0.187753098$$

$$x=\frac{\sqrt{855-15\sqrt{3189}}}{15}\approx 0.187753098$$

$$x = \frac{\sqrt{15 \sqrt{3189} + 855}}{15} \approx 2.750408838$$

$$x = -\frac{\sqrt{15 \sqrt{3189} + 855}}{15} \approx -2.750408838$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$15t^{2}-114t+4=0$$

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 15 for a, -114 for b, and 4 for c in the quadratic formula.

$$t=\frac{-\left(-114\right)±\sqrt{\left(-114\right)^{2}-4\times 15\times 4}}{2\times 15}$$

Do the calculations.

$$t=\frac{114±2\sqrt{3189}}{30}$$

Solve the equation t=\frac{114±2\sqrt{3189}}{30} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{3189}}{15}+\frac{19}{5} t=-\frac{\sqrt{3189}}{15}+\frac{19}{5}$$

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

$$x=\sqrt{\frac{\sqrt{3189}}{15}+\frac{19}{5}} x=-\sqrt{\frac{\sqrt{3189}}{15}+\frac{19}{5}} x=\sqrt{-\frac{\sqrt{3189}}{15}+\frac{19}{5}} x=-\sqrt{-\frac{\sqrt{3189}}{15}+\frac{19}{5}}$$