The roots of \(ax^2 + bx + c = 0\) are real and positive. \(a\), \(b\) and…
2014
The roots of \(ax^2 + bx + c = 0\) are real and positive. \(a\), \(b\) and \(c\) are real. Then \(ax^2 + b|x| + c = 0\) has
- A.
no roots
- B.
2 real roots
- C.
3 real roots
- D.
4 real roots
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Correct answer: D
Solution:
Let the roots of ax^2 + bx + c = 0 be r1 and r2, with r1 > 0 and r2 > 0.
The equation ax^2 + b|x| + c = 0 splits by sign of x:
For x ≥ 0, |x| = x, so the equation is ax^2 + b x + c = 0. Its roots are r1 and r2 (both > 0), so r1 and r2 are solutions of the modified equation.
For x < 0, write x = −y with y > 0. Then ax^2 − b x + c = 0 becomes a y^2 + b y + c = 0. Since y = r1 and y = r2 satisfy a y^2 + b y + c = 0, we get x = −r1 and x = −r2 as solutions (both negative).
Therefore, when r1 and r2 are distinct positive numbers, the modified equation has four distinct real roots: r1, r2, −r1, and −r2. If r1 = r2 = r (a repeated positive root), the modified equation has the two distinct real roots r and −r (each possibly of higher multiplicity).
Hence, in the general (distinct roots) case the equation has four real roots; in the repeated-root special case it has two distinct real roots.