Đặt \(A=a^2+ab+b^2-3a-3b+1989\)

\(\Rightarrow4A=4a^2+4ab+4b^2-12a-12b+7956\)

\(\Rightarrow4A=\left(4a^2+4ab+b^2\right)-\left(12a+6b\right)+9+\left(3b^2-6b+3\right)+7944\)

\(\Rightarrow4A=\left(2a+b\right)^2-6\left(2a+b\right)+9+3\left(b^2-2b+1\right)+7944\)

\(\Rightarrow4A=\left(2a+b-3\right)^2+3\left(b-1\right)^2+7944\)

\(\Rightarrow A=\frac{\left(2a+b-3\right)^2}{4}+\frac{3\left(b-1\right)^2}{4}+1986\)

Vì \(\frac{\left(2a+b-3\right)^2}{4}\ge0\)\(\forall a,b\);  \(\frac{3\left(b-1\right)^2}{4}\ge0\)\(\forall b\)

\(\Rightarrow\frac{\left(2a+b-3\right)^2}{4}+\frac{3\left(b-1\right)^2}{4}\ge0\)\(\forall a,b\)

\(\Rightarrow\frac{\left(2a+b-3\right)^2}{4}+\frac{3\left(b-1\right)^2}{4}+1986\ge1986\)\(\forall a,b\)

\(\Rightarrow A\ge1986\)\(\forall a,b\)

Dấu " = " xảy ra \(\Leftrightarrow\hept{\begin{cases}\frac{\left(2a+b-3\right)^2}{4}=0\\\frac{3\left(b-1\right)^2}{4}=0\end{cases}}\Leftrightarrow\hept{\begin{cases}\left(2a+b-3\right)^2=0\\3\left(b-1\right)^2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}2a+b-3=0\\b-1=0\end{cases}}\) \(\Leftrightarrow\hept{\begin{cases}2a+b-3=0\\b=1\end{cases}}\Leftrightarrow\hept{\begin{cases}2a+1-3=0\\b=1\end{cases}}\Leftrightarrow\hept{\begin{cases}a=1\\b=1\end{cases}}\Leftrightarrow a=b=1\)

Vậy.....

Ta  có : 99ⁿ  + 99^{n + 1} = 99ⁿ + 99ⁿ.99 = 99ⁿ(99 + 1) = 99ⁿ . 100\(⋮\)100 (đpcm)

P = 2x² + 12x + 9

= 2x² + 12x + 18 - 9

= 2( x² + 6x + 9 ) - 9

= 2( x + 3 )² - 9 ≥ -9 ∀ x

Dấu "=" xảy ra khi x = -3

=> MinP = -9 <=> x = -3

\(P=2x^2+12x+9=2\left(x^2+6x+9\right)-9\)

\(=2\left(x+3\right)^2-9\)

Vì \(2\left(x+3\right)^2\ge0\forall x;2\left(x+3\right)^2-9\ge-9\forall x\)

Vậy GTNN là -9 <=> x + 3 = 0 <=> x = -3

banj vieets laij deef ddi

\(\frac{1-4x^2}{x^2+4x}\div\frac{2-4x}{3x}\)

ĐKXĐ : \(\hept{\begin{cases}x\ne0\\x\ne-4\\x\ne\frac{1}{2}\end{cases}}\)

\(=\frac{\left(1-2x\right)\left(1+2x\right)}{x\left(x+4\right)}\times\frac{3x}{2\left(1-2x\right)}\)

\(=\frac{3x\left(1+2x\right)}{2x\left(x+4\right)}\)

\(=\frac{3\left(1+2x\right)}{2\left(x+4\right)}=\frac{6x+3}{2x+8}\)