\(\text{Tử }=\left(x^2-8x+16\right)+\left(x-2-2\sqrt{x-2}.\sqrt{2}+2\right)+2022\)

\(=\left(x-4\right)^2+\left(\sqrt{x-2}-\sqrt{2}\right)^2+2022\ge2022\)

Dấu "=" xảy ra khi \(x-4=0\text{ và }\sqrt{x-2}=\sqrt{2}\Leftrightarrow x=4\).

\(\text{Mẫu }=\sqrt{x-3}+\sqrt{5-x}\)

Ta có: \(\left(a-b\right)^2\ge0\Rightarrow a^2+b^2-2ab\ge0\Rightarrow2\left(a^2+b^2\right)\ge a^2+b^2+2ab=\left(a+b\right)^2\)

Dấu "=" xảy ra khi a = b.

\(\Rightarrow\left(\sqrt{x-3}+\sqrt{5-x}\right)^2\le2\left(x-3+5-x\right)=4\)

\(\Rightarrow\sqrt{x-3}+\sqrt{5-x}\le2\)

Dấu "=" xả ra khi \(\sqrt{x-3}=\sqrt{5-x}\Leftrightarrow x=4\)

\(\Rightarrow Q\ge\frac{2022}{2}=1011\)

Dấu "=" xảy ra khi x = 4.

Vậy GTNN của Q là 1011 khi x = 4.

a/ 4x(7x - 5) - 7x(4x - 2) = -12

  => 28x² - 20x - 28x² + 14x = -12

  => -6x = -12

  => x = 2

b/ (x + 3)(x - 2) + 3x = 4(x + 3/4) 

  => x² + x - 6 + 3x - 4x - 3 = 0

  => x² - 9 = 0

  => x² = 9

  => x = 3 hoặc x = -3

\(B=3x^2+3x-1\)

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

\(=3\left(x^2+x+\dfrac{1}{4}-\dfrac{7}{12}\right)\)

\(=3\left(x+\dfrac{1}{2}\right)^2-\dfrac{7}{4}>=-\dfrac{7}{4}\forall x\)

Dấu '=' xảy ra khi x+1/2=0

=>\(x=-\dfrac{1}{2}\)

\(C=-2x^2+7x+3\)

\(=-2\left(x^2-\dfrac{7}{2}x-\dfrac{3}{2}\right)\)

\(=-2\left(x^2-2\cdot x\cdot\dfrac{7}{4}+\dfrac{49}{16}-\dfrac{73}{16}\right)\)

\(=-2\left(x-\dfrac{7}{4}\right)^2+\dfrac{73}{8}< =\dfrac{73}{8}\forall x\)

Dấu '=' xảy ra khi x-7/4=0

=>x=7/4

\(A=x^2-6x-4=x^2-6x+9-13=\left(x-3\right)^2-13\ge-13\)

Vậy \(A_{min}=-13\Leftrightarrow x=3\)

\(B=x^2-x+1=x^2-2.\frac{1}{2}x+\frac{1}{4}+\frac{3}{4}=\left(x-\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\)

Vậy \(B_{min}=\frac{3}{4}\Leftrightarrow x=\frac{1}{2}\)

\(C=5x^2+x-3=5\left(x^2+\frac{1}{5}x-\frac{3}{5}\right)\)

\(=5\left(x^2+2.\frac{1}{10}x+\frac{1}{100}-\frac{61}{100}\right)\)

\(=5\left[\left(x+\frac{1}{10}\right)^2-\frac{61}{100}\right]=5\left(x+\frac{1}{10}\right)^2-\frac{61}{20}\ge\frac{-61}{20}\)

Vậy \(C_{min}=\frac{-61}{20}\Leftrightarrow x=\frac{-1}{10}\)