Question

Chứng minh hàng đẳng thức sau: a)(a^2-b^2)^2+4(ab)^2=(a^2+b^2)^2 b)(a^2+b^2).(x^2+y^2)=(ax+by)^2

Answer 1

\(a,\left(a^2-b^2\right)^2+4\left(ab\right)^2=a^4-2a^2b^2+b^4+4a^2b^2\\ =a^4+2a^2b^2+b^4=\left(a^2+b^2\right)^2\\ b,\left(a^2+b^2\right)\left(x^2+y^2\right)\\ =a^2x^2+a^2y^2+b^2x^2+b^2y^2\\ \left(ax+by\right)^2=a^2x^2+2axby+b^2y^2\\ \Rightarrow\left(a^2+b^2\right)\left(x^2+y^2\right)\ne\left(ax+by\right)^2\)

Hoặc áp dụng BĐT Bunhiacopski:

\(\left(a^2+b^2\right)\left(x^2+y^2\right)\ge\left(ax+by\right)^2\)

Dấu \("="\Leftrightarrow\dfrac{a}{x}=\dfrac{b}{y}\)