Solve 19^2+n^2=21^2

$$19 ^ { 2 } + n ^ { 2 } = 21 ^ { 2 }$$

Solve for n

$$n=4\sqrt{5}\approx 8.94427191$$

$$n=-4\sqrt{5}\approx -8.94427191$$

Steps by Finding Square Root

Calculate 19 to the power of 2 and get 361.

$$361+n^{2}=21^{2}$$

Calculate 21 to the power of 2 and get 441.

$$361+n^{2}=441$$

Subtract 361 from both sides.

$$n^{2}=441-361$$

Subtract 361 from 441 to get 80.

$$n^{2}=80$$

Take the square root of both sides of the equation.

$$n=4\sqrt{5} n=-4\sqrt{5}$$

Steps Using the Quadratic Formula

Calculate 19 to the power of 2 and get 361.

$$361+n^{2}=21^{2}$$

Calculate 21 to the power of 2 and get 441.

$$361+n^{2}=441$$

Subtract 441 from both sides.

$$361+n^{2}-441=0$$

Subtract 441 from 361 to get -80.

$$-80+n^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$n^{2}-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\left(-80\right)}}{2}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-80\right)}}{2}$$

Multiply -4 times -80.

$$n=\frac{0±\sqrt{320}}{2}$$

Take the square root of 320.

$$n=\frac{0±8\sqrt{5}}{2}$$

Now solve the equation n=\frac{0±8\sqrt{5}}{2} when ± is plus.

$$n=4\sqrt{5}$$

Now solve the equation n=\frac{0±8\sqrt{5}}{2} when ± is minus.

$$n=-4\sqrt{5}$$

The equation is now solved.

$$n=4\sqrt{5} n=-4\sqrt{5}$$