Analyzing Initial Conditions: Roots of \(ax^2 + bx + c = 0\)

The problem states that the quadratic equation \(ax^2 + bx + c = 0\) has roots that are real and positive. Let these roots be \(\alpha\) and \(\beta\). This condition implies several key properties:

• Real Roots: The discriminant must be non-negative. \[ D = b^2 - 4ac \ge 0 \]
• Positive Roots: Both roots must be greater than zero.
  • Sum of roots: \( \alpha + \beta = -\frac{b}{a} > 0 \)
  • Product of roots: \( \alpha \beta = \frac{c}{a} > 0 \)

From these conditions, we deduce:

• \( \frac{c}{a} > 0 \) implies that \(a\) and \(c\) must have the same sign (both positive or both negative).
• \( -\frac{b}{a} > 0 \) implies that \(a\) and \(b\) must have opposite signs.
• Combined, \(a\) and \(c\) have the same sign, and \(b\) has the opposite sign to \(a\) (and \(c\)).
• Also, the condition \( b^2 - 4ac \ge 0 \) must hold.

Investigating the Equation with Absolute Value: \(ax^2 + b|x| + c = 0\)

We need to determine the number of real roots for the equation \(ax^2 + b|x| + c = 0\). Notice that since the original roots \(\alpha, \beta\) are positive, \(c/a > 0\), which means \(c \neq 0\). Therefore, \(x=0\) cannot be a root of \(ax^2 + b|x| + c = 0\) (as it would imply \(c=0\)).

This equation involves \(|x|\). Since \(x^2 = |x|^2\), the equation can be thought of as \(a|x|^2 + b|x| + c = 0\). This structure implies the equation is symmetric with respect to the y-axis, meaning if \(x_0\) is a root, then \(-x_0\) must also be a root. Since \(x=0\) is not a root, all roots must come in pairs of the form \((x_0, -x_0)\) where \(x_0 \neq 0\).

Case Analysis Based on the Sign of \(x\)

We can solve the equation by considering two cases for the absolute value function \(|x|\):

Case 1: \(x \ge 0\)

When \(x \ge 0\), we have \(|x| = x\). The equation simplifies to:

\[ ax^2 + bx + c = 0 \]

The roots of this equation are precisely the original roots \(\alpha\) and \(\beta\). Since we are given that \(\alpha\) and \(\beta\) are positive real numbers, they naturally satisfy the condition \(x \ge 0\). Thus, \(\alpha\) and \(\beta\) are valid roots derived from this case.

Case 2: \(x < 0\)

When \(x < 0\), we have \(|x| = -x\). Substituting this into the equation yields:

\[ ax^2 + b(-x) + c = 0 \] \[ ax^2 - bx + c = 0 \]

Let's analyze the roots of this related quadratic equation, denoted as \(\gamma\) and \(\delta\).

• Sum of roots: \( \gamma + \delta = -\frac{-b}{a} = \frac{b}{a} \)
• Product of roots: \( \gamma \delta = \frac{c}{a} \)

Recall from the initial conditions that \( \frac{c}{a} > 0 \) and \( \frac{b}{a} < 0 \). Applying these to the sum and product of \(\gamma\) and \(\delta\):

• The product \( \gamma \delta = \frac{c}{a} \) is positive, meaning \(\gamma\) and \(\delta\) must have the same sign.
• The sum \( \gamma + \delta = \frac{b}{a} \) is negative, meaning the sum of the roots is negative.

For both the product to be positive and the sum to be negative, both roots \(\gamma\) and \(\delta\) must be negative. Since these roots are negative, they satisfy the condition \(x < 0\). Thus, \(\gamma\) and \(\delta\) are valid roots derived from this case.

An interesting observation is that the roots \(\gamma, \delta\) are the exact negatives of the original roots \(\alpha, \beta\). If we substitute \(y = -x\) (implying \(y > 0\) since \(x < 0\)) into \(ax^2 - bx + c = 0\), we get \(a(-y)^2 - b(-y) + c = 0\), which simplifies to \(ay^2 + by + c = 0\). The positive roots of this equation are known to be \(\alpha\) and \(\beta\). Therefore, the negative roots \(\gamma, \delta\) must be \(-\alpha\) and \(-\beta\).

Combining Roots and Final Count

The set of all real roots for the equation \(ax^2 + b|x| + c = 0\) is the union of the roots found in Case 1 and Case 2:

• Roots from Case 1 (\(x \ge 0\)): \(\alpha, \beta\)
• Roots from Case 2 (\(x < 0\)): \(-\alpha, -\beta\)

The complete set of roots is therefore \(\{ \alpha, \beta, -\alpha, -\beta \}\).

To determine the total number of real roots, we consider the distinctness:

• If the original roots \(\alpha\) and \(\beta\) are distinct positive numbers (\(\alpha \neq \beta\)), then the four roots \(\alpha, \beta, -\alpha, -\beta\) are all distinct real numbers.
• If the original roots are equal positive numbers (\(\alpha = \beta\)), then the roots become \(\alpha, \alpha, -\alpha, -\alpha\). In this specific scenario, there are two distinct real roots: \(\alpha\) and \(-\alpha\).

Considering the general case where the initial roots are distinct, the equation \(ax^2 + b|x| + c = 0\) yields 4 real roots.