The equation is $$x^2–4|x|+3=|k−1|$$ There are several ways to find the solution using either graph or analytically. I want to know is how to do the graphical solution free hand without a calculator. Also in the analytical method I am unable to understand why the product and sum of roots in this equation should be greater than 0. The answer is (-2,4).
Please help.
The equation is equivalent to $(|x|-2)^2 = 1+|k-1|$ and it has at least two distinct real roots, corresponding to $|x| = 2+\sqrt{1+|k-1|}$. The equation has four distinct real roots if and only if $|x|= 2- \sqrt{1+|k-1|}$ also has two distinct real roots, and that happens if and only if $2-\sqrt{1+|k-1|}>0$, or, equivalently, if $|k-1| < 3$ - leading to $k \in (-2, 4)$.