a.
Ta co:
\(\orbr{\begin{cases}x^2-2x-3=0\left(1\right)\left(x\ge0\right)\\x^2+2x-3=0\left(2\right)\left(x< 0\right)\end{cases}}\)
(1)\(\Leftrightarrow\left(x+1\right)\left(x-3\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=-1\left(l\right)\\x=3\left(n\right)\end{cases}}\)
(2)\(\Leftrightarrow\left(x-1\right)\left(x+3\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=1\left(l\right)\\x=-3\left(n\right)\end{cases}}\)
b.
Ta lai co:
\(\orbr{\begin{cases}x^2-2x+1-4a^2=0\left(3\right)\left(x\ge0\right)\\x^2+2x+1-4a^2=0\left(4\right)\left(x< 0\right)\end{cases}}\)
Xet (3)
De phuong trinh dau co 4 nghiem thi PT(3) co nghiem
\(\Rightarrow\Delta^`>0\)
\(\Leftrightarrow4a^2>0\)
\(\Leftrightarrow a>0\)
\(\Rightarrow x_1=1+2a;x_2=1-2a\)
Tuong tu
(4)
\(a>0\)
\(\Rightarrow x_3=-1+2a;x_4=-1-2a\)
\(\Rightarrow S=\left(1+2a\right)^2+\left(1-2a\right)^2+\left(-1+2a\right)^2+\left(-1-2a\right)^2\)
\(=2\left(1+2a\right)^2+2\left(1-2a\right)^2\)
\(\Rightarrow S< +\infty\)
