Solve 98/101=9n^2-9+2/9n^2-1

$$\frac { 98 } { 101 } = \frac { 9 n ^ { 2 } - 9 + 2 } { 9 n ^ { 2 } - 1 }$$

Solve for n

$$n = \frac{\sqrt{203}}{3} \approx 4.74926895$$

$$n = -\frac{\sqrt{203}}{3} \approx -4.74926895$$

Steps by Finding Square Root

Variable n cannot be equal to any of the values -\frac{1}{3},\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 101\left(3n-1\right)\left(3n+1\right), the least common multiple of 101,9n^{2}-1.

$$98\left(3n-1\right)\left(3n+1\right)=101\left(9n^{2}-9+2\right)$$

Use the distributive property to multiply 98 by 3n-1.

$$\left(294n-98\right)\left(3n+1\right)=101\left(9n^{2}-9+2\right)$$

Use the distributive property to multiply 294n-98 by 3n+1 and combine like terms.

$$882n^{2}-98=101\left(9n^{2}-9+2\right)$$

Add -9 and 2 to get -7.

$$882n^{2}-98=101\left(9n^{2}-7\right)$$

Use the distributive property to multiply 101 by 9n^{2}-7.

$$882n^{2}-98=909n^{2}-707$$

Subtract 909n^{2} from both sides.

$$882n^{2}-98-909n^{2}=-707$$

Combine 882n^{2} and -909n^{2} to get -27n^{2}.

$$-27n^{2}-98=-707$$

Add 98 to both sides.

$$-27n^{2}=-707+98$$

Add -707 and 98 to get -609.

$$-27n^{2}=-609$$

Divide both sides by -27.

$$n^{2}=\frac{-609}{-27}$$

Reduce the fraction \frac{-609}{-27} to lowest terms by extracting and canceling out -3.

$$n^{2}=\frac{203}{9}$$

Take the square root of both sides of the equation.

$$n=\frac{\sqrt{203}}{3} n=-\frac{\sqrt{203}}{3}$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values -\frac{1}{3},\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 101\left(3n-1\right)\left(3n+1\right), the least common multiple of 101,9n^{2}-1.

$$98\left(3n-1\right)\left(3n+1\right)=101\left(9n^{2}-9+2\right)$$

Use the distributive property to multiply 98 by 3n-1.

$$\left(294n-98\right)\left(3n+1\right)=101\left(9n^{2}-9+2\right)$$

Use the distributive property to multiply 294n-98 by 3n+1 and combine like terms.

$$882n^{2}-98=101\left(9n^{2}-9+2\right)$$

Add -9 and 2 to get -7.

$$882n^{2}-98=101\left(9n^{2}-7\right)$$

Use the distributive property to multiply 101 by 9n^{2}-7.

$$882n^{2}-98=909n^{2}-707$$

Subtract 909n^{2} from both sides.

$$882n^{2}-98-909n^{2}=-707$$

Combine 882n^{2} and -909n^{2} to get -27n^{2}.

$$-27n^{2}-98=-707$$

Add 707 to both sides.

$$-27n^{2}-98+707=0$$

Add -98 and 707 to get 609.

$$-27n^{2}+609=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -27 for a, 0 for b, and 609 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\left(-27\right)\times 609}}{2\left(-27\right)}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-27\right)\times 609}}{2\left(-27\right)}$$

Multiply -4 times -27.

$$n=\frac{0±\sqrt{108\times 609}}{2\left(-27\right)}$$

Multiply 108 times 609.

$$n=\frac{0±\sqrt{65772}}{2\left(-27\right)}$$

Take the square root of 65772.

$$n=\frac{0±18\sqrt{203}}{2\left(-27\right)}$$

Multiply 2 times -27.

$$n=\frac{0±18\sqrt{203}}{-54}$$

Now solve the equation n=\frac{0±18\sqrt{203}}{-54} when ± is plus.

$$n=-\frac{\sqrt{203}}{3}$$

Now solve the equation n=\frac{0±18\sqrt{203}}{-54} when ± is minus.

$$n=\frac{\sqrt{203}}{3}$$

The equation is now solved.

$$n=-\frac{\sqrt{203}}{3} n=\frac{\sqrt{203}}{3}$$