Problem 25 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If both the roots of \(a x^{2}+b x+c=0\) are positive, then (a) \(-\frac{b}{a}>0\) (b) \(\frac{c}{a}>0\) (c) \(b^{2} \geq 4 a c\) (d) \(a c>0\)

Step-by-Step Solution

Answer

a) \(-\frac{b}{a} > 0\), b) \(\frac{c}{a} > 0\), c) \(b^2 \geq 4ac\), d) \(ac > 0\) are all correct conditions.

Step 1: Understanding the Conditions

For a quadratic equation \( ax^2 + bx + c = 0 \) to have both roots positive, we need two key criteria: the sum of the roots and the product of the roots must both be positive.

Step 2: Applying the Sum of Roots Criterion

The sum of the roots is given by \( -\frac{b}{a} \). For both roots to be positive, this must be positive: \(-\frac{b}{a} > 0\). This means \(\frac{b}{a} < 0\).

Step 3: Applying the Product of Roots Criterion

The product of the roots is given by \( \frac{c}{a} \). For both roots to be positive, this product must be positive: \( \frac{c}{a} > 0 \).

Step 4: Analyzing the Discriminant Condition

The discriminant \( b^2 - 4ac \) must be non-negative for real roots, giving the condition \( b^2 \geq 4ac \).

Step 5: Interpreting the Options

The options provided relate to these conditions:(a) \(-\frac{b}{a} > 0\) implies \(\frac{b}{a} < 0\)(b) \(\frac{c}{a} > 0\) correctly expresses the product of roots condition(c) \( b^2 \geq 4ac \) is required for real roots(d) \( ac > 0 \) also implies \( a \) and \( c \) have the same sign, reinforcing that \( \frac{c}{a} > 0 \).

Key Concepts

Roots of Quadratic Equations

Quadratic equations are central to algebra and appear in the form of \( ax^2 + bx + c = 0 \). The solutions to this equation are known as the roots. Finding these roots involves identifying values of \( x \) that satisfy the equation. The roots can be real or complex, depending on the discriminant. Understanding the nature of these roots is crucial as it helps in analyzing the behavior of the quadratic function.When a quadratic equation has two positive roots:

• Both solutions to the equation are numbers greater than zero.
• This condition significantly affects the values of the coefficients \( a, b, \) and \( c \).

The positivity of roots lays the groundwork for further conditions concerning the sum and the product of these roots.

Sum and Product of Roots

The relationships involving the roots of a quadratic equation can be represented by simple expressions. According to Vieta's formulas:

• The sum of the roots \( \alpha + \beta \) is equal to \( -\frac{b}{a} \).
• The product of the roots \( \alpha \beta \) is equal to \( \frac{c}{a} \).

For both roots to be positive:

• The sum must be positive, meaning \( -\frac{b}{a} > 0 \) or equivalently \( \frac{b}{a} < 0 \). This indicates that \( b \) and \( a \) must have opposite signs.
• The product must also be positive, meaning \( \frac{c}{a} > 0 \). This implies that \( c \) and \( a \) must have the same sign.

These conditions ensure that \( a \), \( b \), and \( c \) interact in such a way that the roots remain positive.

Discriminant Condition

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is defined as \( b^2 - 4ac \). It serves as a critical indicator of the nature of the roots:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real double root.
• If \( b^2 - 4ac < 0 \), the roots are complex and not real.

For both roots to be positive and thus real, the discriminant must be non-negative, ensuring the roots are real numbers:

• Thus, \( b^2 \geq 4ac \) becomes an essential condition in guaranteeing the existence of real roots.

This condition safeguards the possibility of having real numbers as solutions to the quadratic equation.

Positive Roots

Having roots that are strictly positive means both solutions of the equation are above zero. This is not possible by chance but arises out of specific relationships between the coefficients of the equation. It's about ensuring:

• The sum of the roots is positive, which gives insight into the predominant sign of the terms in the equation.
• The product of the roots is positive, indicating balance between the direction and magnitude of the function's curve.

When both roots are positive:

• The relative magnitudes and signs of \( b \) and \( a \), as well as \( c \), work together to position the quadratic function's vertex above the x-axis.

This reflects a harmonious configuration of the coefficients, achieving the desired positivity in roots.

Sign of Coefficients in Quadratics

The signs of the coefficients in the quadratic equation \( ax^2 + bx + c = 0 \) play a crucial role in determining the nature of its roots. For both roots to be positive:

• Since \( \frac{b}{a} < 0 \), \( b \) and \( a \) have opposite signs. This affects the curvature and the slope at which the parabola representing the function rises or falls.
• With \( \frac{c}{a} > 0 \), \( c \) and \( a \) share the same sign. This condition ensures the parabola opens upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)), but roots remain on the positive side of the x-axis.

Such a careful alignment of signs is crucial for maintaining the quadratic's characteristic properties, specifically ensuring the positivity of both roots.