Solve 20.46=n^2-1/n^2+2*76.5/0.9

$$20.46 = \frac { n ^ { 2 } - 1 } { n ^ { 2 } + 2 } \times \frac { 76.5 } { 0.9 }$$

Solve for n

$$n = \frac{2 \sqrt{5079298}}{3227} \approx 1.39679566$$

$$n = -\frac{2 \sqrt{5079298}}{3227} \approx -1.39679566$$

Steps by Finding Square Root

Multiply both sides of the equation by n^{2}+2.

$$20.46\left(n^{2}+2\right)=\left(n^{2}-1\right)\times \frac{76.5}{0.9}$$

Use the distributive property to multiply 20.46 by n^{2}+2.

$$20.46n^{2}+40.92=\left(n^{2}-1\right)\times \frac{76.5}{0.9}$$

Expand \frac{76.5}{0.9} by multiplying both numerator and the denominator by 10.

$$20.46n^{2}+40.92=\left(n^{2}-1\right)\times \frac{765}{9}$$

Divide 765 by 9 to get 85.

$$20.46n^{2}+40.92=\left(n^{2}-1\right)\times 85$$

Use the distributive property to multiply n^{2}-1 by 85.

$$20.46n^{2}+40.92=85n^{2}-85$$

Subtract 85n^{2} from both sides.

$$20.46n^{2}+40.92-85n^{2}=-85$$

Combine 20.46n^{2} and -85n^{2} to get -64.54n^{2}.

$$-64.54n^{2}+40.92=-85$$

Subtract 40.92 from both sides.

$$-64.54n^{2}=-85-40.92$$

Subtract 40.92 from -85 to get -125.92.

$$-64.54n^{2}=-125.92$$

Divide both sides by -64.54.

$$n^{2}=\frac{-125.92}{-64.54}$$

Expand \frac{-125.92}{-64.54} by multiplying both numerator and the denominator by 100.

$$n^{2}=\frac{-12592}{-6454}$$

Reduce the fraction \frac{-12592}{-6454} to lowest terms by extracting and canceling out -2.

$$n^{2}=\frac{6296}{3227}$$

Take the square root of both sides of the equation.

$$n=\frac{2\sqrt{5079298}}{3227} n=-\frac{2\sqrt{5079298}}{3227}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by n^{2}+2.

$$20.46\left(n^{2}+2\right)=\left(n^{2}-1\right)\times \frac{76.5}{0.9}$$

Use the distributive property to multiply 20.46 by n^{2}+2.

$$20.46n^{2}+40.92=\left(n^{2}-1\right)\times \frac{76.5}{0.9}$$

Expand \frac{76.5}{0.9} by multiplying both numerator and the denominator by 10.

$$20.46n^{2}+40.92=\left(n^{2}-1\right)\times \frac{765}{9}$$

Divide 765 by 9 to get 85.

$$20.46n^{2}+40.92=\left(n^{2}-1\right)\times 85$$

Use the distributive property to multiply n^{2}-1 by 85.

$$20.46n^{2}+40.92=85n^{2}-85$$

Subtract 85n^{2} from both sides.

$$20.46n^{2}+40.92-85n^{2}=-85$$

Combine 20.46n^{2} and -85n^{2} to get -64.54n^{2}.

$$-64.54n^{2}+40.92=-85$$

Add 85 to both sides.

$$-64.54n^{2}+40.92+85=0$$

Add 40.92 and 85 to get 125.92.

$$-64.54n^{2}+125.92=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -64.54 for a, 0 for b, and 125.92 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\left(-64.54\right)\times 125.92}}{2\left(-64.54\right)}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-64.54\right)\times 125.92}}{2\left(-64.54\right)}$$

Multiply -4 times -64.54.

$$n=\frac{0±\sqrt{258.16\times 125.92}}{2\left(-64.54\right)}$$

Multiply 258.16 times 125.92 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$n=\frac{0±\sqrt{32507.5072}}{2\left(-64.54\right)}$$

Take the square root of 32507.5072.

$$n=\frac{0±\frac{2\sqrt{5079298}}{25}}{2\left(-64.54\right)}$$

Multiply 2 times -64.54.

$$n=\frac{0±\frac{2\sqrt{5079298}}{25}}{-129.08}$$

Now solve the equation n=\frac{0±\frac{2\sqrt{5079298}}{25}}{-129.08} when ± is plus.

$$n=-\frac{2\sqrt{5079298}}{3227}$$

Now solve the equation n=\frac{0±\frac{2\sqrt{5079298}}{25}}{-129.08} when ± is minus.

$$n=\frac{2\sqrt{5079298}}{3227}$$

The equation is now solved.

$$n=-\frac{2\sqrt{5079298}}{3227} n=\frac{2\sqrt{5079298}}{3227}$$