Solve /n^2=11^2-107^2+96^2+{59}^2

$$\frac{ }{ } { n }^{ 2 } = { 11 }^{ 2 } - { 107 }^{ 2 } + { 96 }^{ 2 } + { 59 }^{ 2 }$$

Solve for n

$$n=-37$$

$$n=37$$

Steps Using Difference of Squares

Anything divided by one gives itself.

$$1n^{2}=11^{2}-107^{2}+96^{2}+59^{2}$$

Calculate 11 to the power of 2 and get 121.

$$1n^{2}=121-107^{2}+96^{2}+59^{2}$$

Calculate 107 to the power of 2 and get 11449.

$$1n^{2}=121-11449+96^{2}+59^{2}$$

Subtract 11449 from 121 to get -11328.

$$1n^{2}=-11328+96^{2}+59^{2}$$

Calculate 96 to the power of 2 and get 9216.

$$1n^{2}=-11328+9216+59^{2}$$

Add -11328 and 9216 to get -2112.

$$1n^{2}=-2112+59^{2}$$

Calculate 59 to the power of 2 and get 3481.

$$1n^{2}=-2112+3481$$

Add -2112 and 3481 to get 1369.

$$1n^{2}=1369$$

Subtract 1369 from both sides.

$$1n^{2}-1369=0$$

Reorder the terms.

$$n^{2}-1369=0$$

Consider n^{2}-1369. Rewrite n^{2}-1369 as n^{2}-37^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(n-37\right)\left(n+37\right)=0$$

To find equation solutions, solve n-37=0 and n+37=0.

$$n=37 n=-37$$

Steps by Finding Square Root

Anything divided by one gives itself.

$$1n^{2}=11^{2}-107^{2}+96^{2}+59^{2}$$

Calculate 11 to the power of 2 and get 121.

$$1n^{2}=121-107^{2}+96^{2}+59^{2}$$

Calculate 107 to the power of 2 and get 11449.

$$1n^{2}=121-11449+96^{2}+59^{2}$$

Subtract 11449 from 121 to get -11328.

$$1n^{2}=-11328+96^{2}+59^{2}$$

Calculate 96 to the power of 2 and get 9216.

$$1n^{2}=-11328+9216+59^{2}$$

Add -11328 and 9216 to get -2112.

$$1n^{2}=-2112+59^{2}$$

Calculate 59 to the power of 2 and get 3481.

$$1n^{2}=-2112+3481$$

Add -2112 and 3481 to get 1369.

$$1n^{2}=1369$$

Divide both sides by 1.

$$n^{2}=1369$$

Take the square root of both sides of the equation.

$$n=37 n=-37$$

Steps Using the Quadratic Formula

Anything divided by one gives itself.

$$1n^{2}=11^{2}-107^{2}+96^{2}+59^{2}$$

Calculate 11 to the power of 2 and get 121.

$$1n^{2}=121-107^{2}+96^{2}+59^{2}$$

Calculate 107 to the power of 2 and get 11449.

$$1n^{2}=121-11449+96^{2}+59^{2}$$

Subtract 11449 from 121 to get -11328.

$$1n^{2}=-11328+96^{2}+59^{2}$$

Calculate 96 to the power of 2 and get 9216.

$$1n^{2}=-11328+9216+59^{2}$$

Add -11328 and 9216 to get -2112.

$$1n^{2}=-2112+59^{2}$$

Calculate 59 to the power of 2 and get 3481.

$$1n^{2}=-2112+3481$$

Add -2112 and 3481 to get 1369.

$$1n^{2}=1369$$

Subtract 1369 from both sides.

$$1n^{2}-1369=0$$

Reorder the terms.

$$n^{2}-1369=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -1369 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\left(-1369\right)}}{2}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-1369\right)}}{2}$$

Multiply -4 times -1369.

$$n=\frac{0±\sqrt{5476}}{2}$$

Take the square root of 5476.

$$n=\frac{0±74}{2}$$

Now solve the equation n=\frac{0±74}{2} when ± is plus. Divide 74 by 2.

$$n=37$$

Now solve the equation n=\frac{0±74}{2} when ± is minus. Divide -74 by 2.

$$n=-37$$

The equation is now solved.

$$n=37 n=-37$$