Question

\(Cho\) \(a^2=b^2+c^2;b^2=2c^2-2013\)

\(CMR:5a^2-7b^2-c^2=4026\)

Answer 1

Từ: \(b^2=2c^2-2013\Rightarrow2c^2-b^2=2013\)

Đặt: \(P=5a^2-7b^2-c^2=5\cdot\left(b^2+c^2\right)-7b^2-c^2=4c^2-2b^2=2\cdot\left(2c^2-b^2\right)=2\cdot2013=4026\)

đpcm.

Answer 2

+ \(a^2=b^2+c^2\Rightarrow5a^2=5b^2+5c^{2.}\) (1)

+ \(b^2=2c^2-2013\Rightarrow7b^2=14c^2-7.2013\Rightarrow7b^2+c^2=15c^2-7.2013\) (2)

+ \(b^2=2c^2-2013\Rightarrow b^2-2c^2=-2013\) (2)

Trừ hai vế của (1) cho hai vế của (2)

\(5a^2-7b^2-c^2=5b^2+5c^2-15c^2+7.2013=5b^2-10c^2+7.2013=\)

\(=5\left(b^2-2c^2\right)+7.2013\) (4)

Thay (3) vào (4)

\(5a^2-7b^2-c^2=5.\left(-2013\right)+7.2013=2.2013=4026\)