Solve a^4+81/4=9a^2

$$a ^ { 4 } + \frac { 81 } { 4 } = 9 a ^ { 2 }$$

Solve for a

$$a = \frac{3 \sqrt{2}}{2} \approx 2.121320344$$

$$a = -\frac{3 \sqrt{2}}{2} \approx -2.121320344$$

Steps Using the Quadratic Formula

Subtract 9a^{2} from both sides.

$$a^{4}+\frac{81}{4}-9a^{2}=0$$

Substitute t for a^{2}.

$$t^{2}-9t+\frac{81}{4}=0$$

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, -9 for b, and \frac{81}{4} for c in the quadratic formula.

$$t=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 1\times \frac{81}{4}}}{2}$$

Do the calculations.

$$t=\frac{9±0}{2}$$

Solutions are the same.

$$t=\frac{9}{2}$$

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

$$a=-\frac{3\sqrt{2}}{2} a=\frac{3\sqrt{2}}{2}$$