Solve A^2+(Asqrt{5})^2=-4^2

$$A ^ { 2 } + ( A \sqrt { 5 } ) ^ { 2 } = - 4 ^ { 2 }$$

Solve for A

$$A=-\frac{2\sqrt{6}i}{3}\approx -0-1.632993162i$$

$$A=\frac{2\sqrt{6}i}{3}\approx 1.632993162i$$

Steps by Finding Square Root

Expand \left(A\sqrt{5}\right)^{2}.

$$A^{2}+A^{2}\left(\sqrt{5}\right)^{2}=-4^{2}$$

The square of \sqrt{5} is 5.

$$A^{2}+A^{2}\times 5=-4^{2}$$

Combine A^{2} and A^{2}\times 5 to get 6A^{2}.

$$6A^{2}=-4^{2}$$

Calculate 4 to the power of 2 and get 16.

$$6A^{2}=-16$$

Divide both sides by 6.

$$A^{2}=\frac{-16}{6}$$

Reduce the fraction \frac{-16}{6} to lowest terms by extracting and canceling out 2.

$$A^{2}=-\frac{8}{3}$$

The equation is now solved.

$$A=\frac{2\sqrt{6}i}{3} A=-\frac{2\sqrt{6}i}{3}$$

Steps Using the Quadratic Formula

Expand \left(A\sqrt{5}\right)^{2}.

$$A^{2}+A^{2}\left(\sqrt{5}\right)^{2}=-4^{2}$$

The square of \sqrt{5} is 5.

$$A^{2}+A^{2}\times 5=-4^{2}$$

Combine A^{2} and A^{2}\times 5 to get 6A^{2}.

$$6A^{2}=-4^{2}$$

Calculate 4 to the power of 2 and get 16.

$$6A^{2}=-16$$

Add 16 to both sides.

$$6A^{2}+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 0 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$A=\frac{0±\sqrt{0^{2}-4\times 6\times 16}}{2\times 6}$$

Square 0.

$$A=\frac{0±\sqrt{-4\times 6\times 16}}{2\times 6}$$

Multiply -4 times 6.

$$A=\frac{0±\sqrt{-24\times 16}}{2\times 6}$$

Multiply -24 times 16.

$$A=\frac{0±\sqrt{-384}}{2\times 6}$$

Take the square root of -384.

$$A=\frac{0±8\sqrt{6}i}{2\times 6}$$

Multiply 2 times 6.

$$A=\frac{0±8\sqrt{6}i}{12}$$

Now solve the equation A=\frac{0±8\sqrt{6}i}{12} when ± is plus.

$$A=\frac{2\sqrt{6}i}{3}$$

Now solve the equation A=\frac{0±8\sqrt{6}i}{12} when ± is minus.

$$A=-\frac{2\sqrt{6}i}{3}$$

The equation is now solved.

$$A=\frac{2\sqrt{6}i}{3} A=-\frac{2\sqrt{6}i}{3}$$