Solve 127/136=1-1/n^2

$$\frac { 127 } { 136 } = 1 - \frac { 1 } { n ^ { 2 } }$$

Solve for n

$$n = \frac{2 \sqrt{34}}{3} \approx 3.887301263$$

$$n = -\frac{2 \sqrt{34}}{3} \approx -3.887301263$$

Steps by Finding Square Root

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 136n^{2}, the least common multiple of 136,n^{2}.

$$127n^{2}=136n^{2}-136$$

Subtract 136n^{2} from both sides.

$$127n^{2}-136n^{2}=-136$$

Combine 127n^{2} and -136n^{2} to get -9n^{2}.

$$-9n^{2}=-136$$

Divide both sides by -9.

$$n^{2}=\frac{-136}{-9}$$

Fraction \frac{-136}{-9} can be simplified to \frac{136}{9} by removing the negative sign from both the numerator and the denominator.

$$n^{2}=\frac{136}{9}$$

Take the square root of both sides of the equation.

$$n=\frac{2\sqrt{34}}{3} n=-\frac{2\sqrt{34}}{3}$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 136n^{2}, the least common multiple of 136,n^{2}.

$$127n^{2}=136n^{2}-136$$

Subtract 136n^{2} from both sides.

$$127n^{2}-136n^{2}=-136$$

Combine 127n^{2} and -136n^{2} to get -9n^{2}.

$$-9n^{2}=-136$$

Add 136 to both sides.

$$-9n^{2}+136=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 0 for b, and 136 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\left(-9\right)\times 136}}{2\left(-9\right)}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-9\right)\times 136}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$n=\frac{0±\sqrt{36\times 136}}{2\left(-9\right)}$$

Multiply 36 times 136.

$$n=\frac{0±\sqrt{4896}}{2\left(-9\right)}$$

Take the square root of 4896.

$$n=\frac{0±12\sqrt{34}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$n=\frac{0±12\sqrt{34}}{-18}$$

Now solve the equation n=\frac{0±12\sqrt{34}}{-18} when ± is plus.

$$n=-\frac{2\sqrt{34}}{3}$$

Now solve the equation n=\frac{0±12\sqrt{34}}{-18} when ± is minus.

$$n=\frac{2\sqrt{34}}{3}$$

The equation is now solved.

$$n=-\frac{2\sqrt{34}}{3} n=\frac{2\sqrt{34}}{3}$$