Find the range of values of k if the equation Xsquare+(4K-2)X+(5Ksqure-4)=0 has 2 real and distinct roots of opposite signs.
Thanks.
How do I type X squre in the text box? Thank.
x^2 is the usual convention.
Note: The latex code for $\displaystyle x^2$ is [tex]x^2[/tex]. See here: http://www.mathhelpforum.com/math-he...-tutorial.html
The roots are real and distinct $\displaystyle \Rightarrow\Delta>0$
$\displaystyle \Delta=b^2-4ac=4(-k^2-4k+5)$
$\displaystyle k_1=-5, \ k_2=1\Rightarrow k\in(-5,1)$ (1)
The roots are of opposite signs$\displaystyle \Rightarrow x_1x_2<0$
$\displaystyle x_1x_2=5k^2-4<0\Rightarrow k\in\left(-\frac{2}{\sqrt{5}},\frac{2}{\sqrt{5}}\right)$ (2)
From (1) and (2)$\displaystyle \Rightarrow x\in\left(-\frac{2}{\sqrt{5}},\frac{2}{\sqrt{5}}\right)$