2.
A = x2 - 4x + 10 = (x2 - 2.x.2 + 22) + 6 = (x - 2)2 + 6 \(\ge\) 6
( do (x - 2)2 \(\ge\) 0)
Vậy: GTNN của A là 6 (tại x = 2)
B = x2 - x + 1 = (x2 - 2.x.\(\frac{1}{2}\) + \(\frac{1}{4}\)) + \(\frac{3}{4}\) = \(\left(x-\frac{1}{2}\right)^2\) + \(\frac{3}{4}\) \(\ge\) \(\frac{3}{4}\)
Vậy: GTNN của B là \(\frac{3}{4}\) (tại x = \(\frac{1}{2}\) )
C = 2x2 - 8x = 2 (x2 - 4x) = 2(x2 - 2.x.2 + 4) - 8 = 2(x - 2)2 - 8 \(\ge\) -8
Vậy : GTNN của C là -8 (tại x = 2)
Bài 1:
a)
\((x-5)(2x-1)-4x(x+2)=-(x-1)^2-2x(x-3)\)
\(\Leftrightarrow (2x^2-11x+5)-(4x^2+8x)=-(x^2-2x+1)-(2x^2-6x)\)
\(\Leftrightarrow -2x^2-19x+5=-3x^2+8x-1\)
\(\Leftrightarrow x^2-27x+6=0\)
\(\Leftrightarrow (x-\frac{27}{2})^2=\frac{705}{4}\Rightarrow \left[\begin{matrix} x-\frac{27}{2}=\frac{\sqrt{705}}{2}\\ x-\frac{27}{2}=\frac{-\sqrt{705}}{2}\end{matrix}\right.\)
\(\Rightarrow \left[\begin{matrix} x=\frac{27+\sqrt{705}}{2}\\ x=\frac{27-\sqrt{705}}{2}\end{matrix}\right.\)
b)
\((4x-1)-(2x+3)^2-12x(x+3)=1\)
\(\Leftrightarrow 4x-1-(4x^2+12x+9)-(12x^2+36x)=1\)
\(\Leftrightarrow -16x^2-44x-11=0\)
\(\Leftrightarrow 16x^2+44x+11=0\)
\(\Leftrightarrow (4x+\frac{11}{2})^2=\frac{77}{4}\)
\(\Rightarrow \left[\begin{matrix} 4x+\frac{11}{2}=\frac{\sqrt{77}}{2}\\ 4x+\frac{11}{2}=\frac{-\sqrt{77}}{2}\end{matrix}\right.\Rightarrow \left[\begin{matrix} x=\frac{\sqrt{77}-11}{8}\\ x=\frac{-\sqrt{77}-11}{8}\end{matrix}\right.\)
c)
\((2x-1)^2-(3x+5)^2=5x(x+3)-24\)
\(\Leftrightarrow (4x^2-4x+1)-(9x^2+30x+25)=5x^2+15x-24\)
\(\Leftrightarrow -5x^2-34x-24=5x^2+15x-24\)
\(\Leftrightarrow 10x^2+49x=0\Leftrightarrow x(10x+49)=0\Rightarrow \left[\begin{matrix} x=0\\ x=\frac{-49}{10}\end{matrix}\right.\)
d)
\((\frac{1}{2}x-3)^2-x(x+3)=(x-1)^2-10\)
\(\Leftrightarrow \frac{1}{4}x^2-3x+9-(x^2+3x)=x^2-2x+1-10\)
\(\Leftrightarrow \frac{-3}{4}x^2-6x+9=x^2-2x-9\)
\(\Leftrightarrow \frac{7}{4}x^2+4x-18=0\)
\(\Leftrightarrow 7x^2+16x-72=0\)
\(\Leftrightarrow 7(x+\frac{8}{7})^2=\frac{568}{7}\)
\(\Rightarrow \left[\begin{matrix} x+\frac{8}{7}=\frac{\sqrt{568}}{7}\\ x+\frac{8}{7}=\frac{-\sqrt{568}}{7}\end{matrix}\right.\Rightarrow \left[\begin{matrix} x=\frac{\sqrt{568}-8}{7}\\ x=\frac{-\sqrt{568}-8}{7}\end{matrix}\right.\)