Solve sqrt{x^4}+5x^2=81

$$\sqrt{ { x }^{ 4 } } +5 { x }^{ 2 } = 81$$

Solve for x

$$x = \frac{3 \sqrt{6}}{2} \approx 3.674234614$$

$$x = -\frac{3 \sqrt{6}}{2} \approx -3.674234614$$

Solution Steps

Subtract 5x^{2} from both sides of the equation.

$$\sqrt{x^{4}}=81-5x^{2}$$

Square both sides of the equation.

$$\left(\sqrt{x^{4}}\right)^{2}=\left(81-5x^{2}\right)^{2}$$

Calculate \sqrt{x^{4}} to the power of 2 and get x^{4}.

$$x^{4}=\left(81-5x^{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(81-5x^{2}\right)^{2}.

$$x^{4}=6561-810x^{2}+25\left(x^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}=6561-810x^{2}+25x^{4}$$

Subtract 6561 from both sides.

$$x^{4}-6561=-810x^{2}+25x^{4}$$

Add 810x^{2} to both sides.

$$x^{4}-6561+810x^{2}=25x^{4}$$

Subtract 25x^{4} from both sides.

$$x^{4}-6561+810x^{2}-25x^{4}=0$$

Combine x^{4} and -25x^{4} to get -24x^{4}.

$$-24x^{4}-6561+810x^{2}=0$$

Substitute t for x^{2}.

$$-24t^{2}+810t-6561=0$$

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, 810 for b, and -6561 for c in the quadratic formula.

$$t=\frac{-810±\sqrt{810^{2}-4\left(-24\right)\left(-6561\right)}}{-24\times 2}$$

Do the calculations.

$$t=\frac{-810±162}{-48}$$

Solve the equation t=\frac{-810±162}{-48} when ± is plus and when ± is minus.

$$t=\frac{27}{2} t=\frac{81}{4}$$

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

$$x=\frac{3\sqrt{6}}{2} x=-\frac{3\sqrt{6}}{2} x=\frac{9}{2} x=-\frac{9}{2}$$

Substitute \frac{3\sqrt{6}}{2} for x in the equation \sqrt{x^{4}}+5x^{2}=81.

$$\sqrt{\left(\frac{3\sqrt{6}}{2}\right)^{4}}+5\times \left(\frac{3\sqrt{6}}{2}\right)^{2}=81$$

Simplify. The value x=\frac{3\sqrt{6}}{2} satisfies the equation.

$$81=81$$

Substitute -\frac{3\sqrt{6}}{2} for x in the equation \sqrt{x^{4}}+5x^{2}=81.

$$\sqrt{\left(-\frac{3\sqrt{6}}{2}\right)^{4}}+5\left(-\frac{3\sqrt{6}}{2}\right)^{2}=81$$

Simplify. The value x=-\frac{3\sqrt{6}}{2} satisfies the equation.

$$81=81$$

Substitute \frac{9}{2} for x in the equation \sqrt{x^{4}}+5x^{2}=81.

$$\sqrt{\left(\frac{9}{2}\right)^{4}}+5\times \left(\frac{9}{2}\right)^{2}=81$$

Simplify. The value x=\frac{9}{2} does not satisfy the equation.

$$\frac{243}{2}=81$$

Substitute -\frac{9}{2} for x in the equation \sqrt{x^{4}}+5x^{2}=81.

$$\sqrt{\left(-\frac{9}{2}\right)^{4}}+5\left(-\frac{9}{2}\right)^{2}=81$$

Simplify. The value x=-\frac{9}{2} does not satisfy the equation.

$$\frac{243}{2}=81$$

List all solutions of \sqrt{x^{4}}=81-5x^{2}.

$$x=\frac{3\sqrt{6}}{2} x=-\frac{3\sqrt{6}}{2}$$