Question

Cho \(\frac{a}{c}=\frac{c}{b}\) . Chứng minh rằng

\(1,\frac{a^2+c^2}{b^2+c^2}=\frac{a}{b}\)

\(2,\frac{b^2-a^2}{a^2+c^2}=\frac{b-a}{a}\)

Bài 2 : Tìm x

\(\left|x^2+\right|6x-2\left|\right|=x^2+4\)

Answer 1

\(\frac{a}{c}=\frac{c}{b}\Rightarrow\frac{a^2}{c^2}=\frac{c^2}{b^2}=\frac{a^2+c^2}{c^2+b^2}\) (1)

Mặt khác \(\frac{a}{c}=\frac{c}{b}\Rightarrow ab=c^2\Rightarrow\frac{ab}{b^2}=\frac{c^2}{b^2}\Rightarrow\frac{a}{b}=\frac{c^2}{b^2}\) (2)

(1);(2) \(\Rightarrow\frac{a}{b}=\frac{a^2+c^2}{b^2+c^2}\)

Từ trên ta cũng có: \(\frac{b^2+c^2}{a^2+c^2}=\frac{b}{a}\)

\(\Rightarrow\frac{b^2+c^2}{a^2+c^2}-1=\frac{b}{a}-1\)

\(\Rightarrow\frac{b^2+c^2-a^2-c^2}{a^2+c^2}=\frac{b-a}{a}\)

\(\Rightarrow\frac{b^2-a^2}{a^2+c^2}=\frac{b-a}{a}\)