Solve 8n^4-42=16n^2

$$8 n ^ { 4 } - 42 = 16 n ^ { 2 }$$

Solve for n

$$n = -\frac{\sqrt{14}}{2} \approx -1.870828693$$

$$n = \frac{\sqrt{14}}{2} \approx 1.870828693$$

Steps Using the Quadratic Formula

Subtract 16n^{2} from both sides.

$$8n^{4}-42-16n^{2}=0$$

Substitute t for n^{2}.

$$8t^{2}-16t-42=0$$

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 8 for a, -16 for b, and -42 for c in the quadratic formula.

$$t=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 8\left(-42\right)}}{2\times 8}$$

Do the calculations.

$$t=\frac{16±40}{16}$$

Solve the equation t=\frac{16±40}{16} when ± is plus and when ± is minus.

$$t=\frac{7}{2} t=-\frac{3}{2}$$

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

$$n=\frac{\sqrt{14}}{2} n=-\frac{\sqrt{14}}{2}$$