Question

[Cold Wind]

Source of Question: Câu hỏi của Hiếu Cao Huy - Toán lớp 9 | Học trực tuyến

Xét pt (1): \(\Delta=b^2-4ac\)

\(x_1=\dfrac{-b+\sqrt{\Delta}}{2a}\); \(x_2=\dfrac{-b-\sqrt{\Delta}}{2a}\)

Xét pt (2) : \(\Delta=b^2-4ac\)

\(y_1=\dfrac{-b+\sqrt{\Delta}}{2c}\) ; \(y_2=\dfrac{-b-\sqrt{\Delta}}{2c}\)

Thay vào M:

\(M=\dfrac{\left(-b+\sqrt{\Delta}\right)^2}{4a^2}+\dfrac{\left(-b-\sqrt{\Delta}\right)^2}{4a^2}+\dfrac{\left(-b+\sqrt{\Delta}\right)^2}{4c^2}+\dfrac{\left(-b-\sqrt{\Delta}\right)^2}{4c^2}\)

\(=\dfrac{b^2-2b\sqrt{\Delta}+\Delta}{4a^2}+\dfrac{b^2+2b\sqrt{\Delta}+\Delta}{4a^2}+\dfrac{b^2-2b\sqrt{\Delta}+\Delta}{4c^2}+\dfrac{b^2+2b\sqrt{\Delta}+\Delta}{4c^2}\)

\(=\dfrac{2b^2+2\Delta}{4a^2}+\dfrac{2b^2+2\Delta}{4c^2}=\dfrac{b^2+\Delta}{2a^2}+\dfrac{b^2+\Delta}{2c^2}=\dfrac{b^2c^2+\Delta c^2}{2a^2c^2}+\dfrac{a^2b^2+\Delta a^2}{2a^2c^2}\)

\(=\dfrac{b^2\left(a^2+c^2\right)+\Delta\left(a^2+c^2\right)}{2a^2c^2}=\dfrac{\left(b^2+\Delta\right)\left(a^2+c^2\right)}{2a^2c^2}=\dfrac{\left(b^2+b^2-4ac\right)\left(a^2+c^2\right)}{2a^2c^2}\)

\(=\dfrac{\left(2b^2-4ac\right)\left(a^2+c^2\right)}{2a^2c^2}=\dfrac{\left(b^2-2ac\right)\left(a^2+c^2\right)}{a^2c^2}=\dfrac{a^2b^2-2a^3c+b^2c^2-2ac^3}{a^2c^2}\)

\(=\dfrac{a^2b^2}{a^2c^2}+\dfrac{b^2c^2}{a^2c^2}-\dfrac{2a^3c}{a^2c^2}-\dfrac{2ac^3}{a^2c^2}=\dfrac{b^2}{c^2}+\dfrac{b^2}{a^2}-\dfrac{2a}{c}-\dfrac{2c}{a}\)

\(=\left(\dfrac{b^2}{c^2}-\dfrac{2ac}{c^2}\right)+\left(\dfrac{b^2}{a^2}-\dfrac{2ac}{a^2}\right)=\dfrac{b^2-2ac}{c^2}+\dfrac{b^2-2ac}{a^2}\)

\(=\left(b^2-2ac\right)\left(\dfrac{1}{c^2}+\dfrac{1}{a^2}\right)\)

Thanks a lots for your answering ^^!

Hiếu Cao Huy: Wait together!

Answer 1

[Neet]

M=\(\left(x_1+x_2\right)^2-2x_1.x_2+\left(y_1+y_2\right)^2-2y_1.y_2\)

Áp dụng định lý viettel :( :v )

\(\left\{{}\begin{matrix}x_1+x_2=-\dfrac{b}{a}\\x_1x_2=\dfrac{c}{a}\end{matrix}\right.\);\(\left\{{}\begin{matrix}y_1+y_2=-\dfrac{b}{c}\\y_1y_2=\dfrac{a}{c}\end{matrix}\right.\)

\(M=\dfrac{b^2}{a^2}-\dfrac{2c}{a}+\dfrac{b^2}{c^2}-\dfrac{2a}{c}=\dfrac{b^2-4ac}{a^2}+\dfrac{b^2-4ac}{c^2}+2\left(\dfrac{a}{c}+\dfrac{c}{a}\right)\)

\(\ge2\left(\dfrac{a}{c}+\dfrac{c}{a}\right)\ge4\)

Dấu = xảy ra: \(\left\{{}\begin{matrix}a=c\\b^2=4ac\end{matrix}\right.\)\(\Leftrightarrow b^2=4a^2=4c^2\)

Answer 2

[Mysterious Person]

@_@ đưa thẳng câu hỏi luôn đi ; nói như zầy chưa nghỉ ra câu trả lời ; chống mặt chết trước rồi