Question

chứng minh các biểu thức sau luôn dương (>0)

12) L=4y-2x+x²

13) M=x²+y²+6x-4y+15

!4)N=3x²-5x+3

16) P=5x²-16x+9

17)Q=x²-4xy+4y²+10

18)R=x²-4xy+5y²-2y+7

19)S=2x²-4xy+5xy²+10x-22y+28

20)T=x²+4y²-2x-4xy+4y+2022

21)V=x²+2y²-2xy+4x-2y+2023

Answer 1

13: \(=x^2+6x+9+y^2-4y+4+2\)

\(=\left(x+3\right)^2+\left(y-2\right)^2+2>0\)

14: \(=3\left(x^2-\dfrac{5}{3}x+1\right)\)

\(=3\left(x^2-2\cdot x\cdot\dfrac{5}{6}+\dfrac{25}{36}+\dfrac{11}{36}\right)\)

\(=3\left(x-\dfrac{5}{6}\right)^2+\dfrac{11}{12}>0\)

15: \(=5\left(x^2-\dfrac{16}{5}x+\dfrac{9}{5}\right)\)

\(=5\left(x^2-2\cdot x\cdot\dfrac{8}{5}+\dfrac{64}{25}+\dfrac{161}{25}\right)\)

\(=5\left(x-\dfrac{8}{5}\right)^2+\dfrac{161}{5}>0\)