3) a) \(x^2+y^2=\left(x-y\right)^2+2xy=\left(-1\right)^2+2.2=5\)
b) \(x^4+y^4=\left(x^2+y^2\right)^2-2\left(xy\right)^2=5^2-2.2^2=17\)
c) \(x^3y+xy^3=xy\left(x^2+y^2\right)=2.5=10\)
1) a) 992+2.99+12=(99+1)2=1002=10000
b) 49.51= (50-1)(50+1)=502-12=2500-1=2499
c) 452-90+5=452-2.45.1+12+4=(45-1)2+4=1940
2) \(A=x^2-10x+6\)
\(A=x^2-2.x.5+5^2-5^2+6\)
\(A=\left(x-5\right)^2-19\)
Ta có: \(\left(x-5\right)^2\ge0\Rightarrow\left(x-5\right)^2-19\ge-19\)
Vậy: \(A\ge-19\)
\(\Rightarrow Min_A=-19\forall x\in R\) khi và chỉ khi \(\left(x-5\right)^2=0\Leftrightarrow x=5\)
b) \(B=9y^2+6y+16\)
\(B=\left(3y\right)^2+2.3y.1+1^2-1^2+16\)
\(B=\left(3y+1\right)^2+15\)
Ta có: \(\left(3y+1\right)^2\ge0\forall x\in R\Rightarrow\left(3y+1\right)^2+15\ge15\)
Vậy: \(A\ge15\Rightarrow Min_A=15\forall x\in R\) khi và chỉ khi:
\(\left(3y+1\right)^2=0\Leftrightarrow y=-\dfrac{1}{3}\)
c) \(C=x^2+y^2+6x-10y+100\)
\(C=x^2+2.x.3+3^2+y^2-2.y.5+5^2-3^2-5^2+100\)
\(C=\left(x+3\right)^2+\left(y-5\right)^2+66\)
Làm tương tự \(\Rightarrow Min_C=66\) khi x=-3 và y=5
