Question

Cho \(z=a+bi\). Chứng minh rằng :

a) \(z^2+\left(\overline{z}\right)^2=2\left(a^2-b^2\right)\)

b) \(z^2-\left(\overline{z}\right)^2=4abi\)

c) \(z^2\left(\overline{z}\right)^2=\left(a^2+b^2\right)^2\)

Answer 1

\(VT=\left(a+bi\right)^2+\left(a-bi\right)^2\\ =a^2+2abi-b^2+a^2-2abi-b^2\\ =2a^2-2b^2\\ =2\left(a^2-b^2\right)=VP\)

\(VT=\left(a+bi\right)^2-\left(a-bi\right)^2\\ =a^2+2abi-b^2-\left(a^2-2abi-b^2\right)\\ =a^2+2abi-b^2-a^2+2abi+b^2\\ =4abi=VP\)

\(VT=\left(a+bi\right)^2\left(a-bi\right)^2\\ =\left[\left(a+bi\right)\left(a-bi\right)\right]^2\\ =\left[a^2-\left(bi\right)^2\right]^2\\ =\left(a^2+b^2\right)^2=VP\)