What will be the value of k for which equation $x^2–4|x|+3=|k−1|$ has four real roots?

Question

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.

score 2 · Answer 1 · answered Mar 20 '23 at 02:11

2

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 $k \in (-2, 4)$.

answered Mar 20 '23 at 02:11

Catalin Zara

6,187

Ok. Could u please tell me why it has to be greater than 0 for it to have two distinct real roots. – SAGNIK UPADHYAY Mar 20 '23 at 02:44
How many roots does $|x| = -1$ have? How about $|x| = 0$ or $|x| = 42$? – Catalin Zara Mar 20 '23 at 03:40
Think about the discriminant. $b^2 - 4ac$. – Bumblebee Mar 20 '23 at 03:41
Ok i get it |x|>= 0. Thnx @Catalin Zara – SAGNIK UPADHYAY Mar 20 '23 at 04:00
But i would still want to know. Why if c/a > 0, then too the answer comes. I can't wrap my head around why in this eqn product and sum of roots is greater than 0. Can u give some insight? – SAGNIK UPADHYAY Mar 20 '23 at 04:01

What will be the value of k for which equation $x^2–4|x|+3=|k−1|$ has four real roots?

1 Answers1