Property 7
In some equations, change of a reliable parameter will result in a quadratic equation.
Example
\(x^2-5|x|+6=0\)
\(x^2=|x|^2\qquad |x|=t\)
\(\Rightarrow t^2-5t+6=0\)
\(9^x-4\cdot3^x+3=0\)
\(3^x=t\)
\(\Rightarrow t^2-4t+3=0\)
\(x^4-5x^2+4=0\)
\(x^2=t\)
\(\Rightarrow t^2-5t+4=0\)
\(\left(\samefrac{x}{x+1}\right)^2-7\left(\samefrac{x}{x+1}\right)+12=0\)
\(\samefrac{x}{x+1}=t\Rightarrow t^2-7t+12=0\)
\(\sum x\): sum of x values
\(\prod x\): Multiplication of x values
S.S. : Solution set
1.
\(x^4-13x^2+36=0\)
\(S.S.=\{x_1,x_2,x_3,x_4\}\)
\(\Rightarrow\sum x=?\)
2.
\(x^4-8x^2+12=0\)
\(S.S.=\{x_1,x_2,x_3,x_4\}\)
\(\Rightarrow\prod x=?\)
3.
\((x-1)^2-(x-1)-12=0\)
\(S.S.=\{x_1,x_2\}\)
\(\Rightarrow x_1+x_2=?\)
4.
\((x+2)^2+(x+2)-6=0\)
\(S.S.=\{x_1,x_2\}\)
\(\Rightarrow x_1x_2=?\)
5.
\(4^x-10\cdot2^x+16=0\)
\(S.S.=\{x_1,x_2\}\)
\(\Rightarrow x_1+x_2=?\)
6.
\(9^x-4\cdot3^x+3=0\)
\(S.S.=\{x_1,x_2\}\)
\(\Rightarrow x_1+x_2=?\)
7.
\((x^2+2x)^2-2(x^2+2x)-3=0\)
\(S.S.=\{x_1,x_2,x_3,x_4\}\)
\(\Rightarrow\prod x=?\)
8.
\(\left(\samefrac{x-1}{x}\right)^2-5\left(\samefrac{x-1}{x}\right)+6=0\)
\(\Rightarrow S.S.=?\)
9.
\(x^2+3|x|-4=0\)
\(\Rightarrow\sum x=?\)
10.
\(x^2-2|x|-3=0\)
\(\Rightarrow\prod x=?\)