Solve n^2-8=113?quadn^2-105

$$n ^ { 2 } - 8 = 113 ? \quad n ^ { 2 } - 105$$

Solve for n

$$n=\frac{\sqrt{679}}{28}\approx 0.930629587$$

$$n=-\frac{\sqrt{679}}{28}\approx -0.930629587$$

Steps by Finding Square Root

Subtract 113n^{2} from both sides.

$$n^{2}-8-113n^{2}=-105$$

Combine n^{2} and -113n^{2} to get -112n^{2}.

$$-112n^{2}-8=-105$$

Add 8 to both sides.

$$-112n^{2}=-105+8$$

Add -105 and 8 to get -97.

$$-112n^{2}=-97$$

Divide both sides by -112.

$$n^{2}=\frac{-97}{-112}$$

Fraction \frac{-97}{-112} can be simplified to \frac{97}{112} by removing the negative sign from both the numerator and the denominator.

$$n^{2}=\frac{97}{112}$$

Take the square root of both sides of the equation.

$$n=\frac{\sqrt{679}}{28} n=-\frac{\sqrt{679}}{28}$$

Steps Using the Quadratic Formula

Subtract 113n^{2} from both sides.

$$n^{2}-8-113n^{2}=-105$$

Combine n^{2} and -113n^{2} to get -112n^{2}.

$$-112n^{2}-8=-105$$

Add 105 to both sides.

$$-112n^{2}-8+105=0$$

Add -8 and 105 to get 97.

$$-112n^{2}+97=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -112 for a, 0 for b, and 97 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\left(-112\right)\times 97}}{2\left(-112\right)}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-112\right)\times 97}}{2\left(-112\right)}$$

Multiply -4 times -112.

$$n=\frac{0±\sqrt{448\times 97}}{2\left(-112\right)}$$

Multiply 448 times 97.

$$n=\frac{0±\sqrt{43456}}{2\left(-112\right)}$$

Take the square root of 43456.

$$n=\frac{0±8\sqrt{679}}{2\left(-112\right)}$$

Multiply 2 times -112.

$$n=\frac{0±8\sqrt{679}}{-224}$$

Now solve the equation n=\frac{0±8\sqrt{679}}{-224} when ± is plus.

$$n=-\frac{\sqrt{679}}{28}$$

Now solve the equation n=\frac{0±8\sqrt{679}}{-224} when ± is minus.

$$n=\frac{\sqrt{679}}{28}$$

The equation is now solved.

$$n=-\frac{\sqrt{679}}{28} n=\frac{\sqrt{679}}{28}$$