Solve x^4-5x^2-75/4=0

$$x ^ { 4 } - 5 x ^ { 2 } - \frac { 75 } { 4 } = 0$$

Solve for x (complex solution)

$$x = \frac{\sqrt{30}}{2} \approx 2.738612788$$

$$x = -\frac{\sqrt{30}}{2} \approx -2.738612788$$

$$x=-\frac{\sqrt{10}i}{2}\approx -0-1.58113883i$$

$$x=\frac{\sqrt{10}i}{2}\approx 1.58113883i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-5t-\frac{75}{4}=0$$

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 1 for a, -5 for b, and -\frac{75}{4} for c in the quadratic formula.

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 1\left(-\frac{75}{4}\right)}}{2}$$

Do the calculations.

$$t=\frac{5±10}{2}$$

Solve the equation t=\frac{5±10}{2} when ± is plus and when ± is minus.

$$t=\frac{15}{2} t=-\frac{5}{2}$$

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

$$x=-\frac{\sqrt{30}}{2} x=\frac{\sqrt{30}}{2} x=-\frac{\sqrt{10}i}{2} x=\frac{\sqrt{10}i}{2}$$

Solve for x

$$x = -\frac{\sqrt{30}}{2} \approx -2.738612788$$

$$x = \frac{\sqrt{30}}{2} \approx 2.738612788$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-5t-\frac{75}{4}=0$$

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 1 for a, -5 for b, and -\frac{75}{4} for c in the quadratic formula.

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 1\left(-\frac{75}{4}\right)}}{2}$$

Do the calculations.

$$t=\frac{5±10}{2}$$

Solve the equation t=\frac{5±10}{2} when ± is plus and when ± is minus.

$$t=\frac{15}{2} t=-\frac{5}{2}$$

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

$$x=\frac{\sqrt{30}}{2} x=-\frac{\sqrt{30}}{2}$$