Multiple choice

If $a,b,c$ are positive real numbers, then the number of real roots of the equation $ ax^{2}+b\left |x \right |+c=0 $ is

  1. $2$
  2. $4$
  3. $0$
  4. $1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since a, b, c are positive, ax^2 + b|x| + c is always positive for any real x. Specifically, if x is real, x^2 >= 0 and |x| >= 0. Thus, ax^2 + b|x| + c >= c > 0. The equation cannot equal zero.

AI explanation

For the equation ax^2 + b|x| + c = 0 where a, b, and c are positive real numbers, we substitute y = |x| where y is non-negative. This transforms the equation to ay^2 + by + c = 0. The discriminant of this quadratic is b^2 - 4ac; because a, b, and c are positive, this value could be positive, but the sum and product of the roots are strictly negative. This means any real roots would be negative, which contradicts the requirement that y is non-negative. Therefore, there are no valid real roots for y, meaning there are no real roots for x. The result is 0.

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