If p, q, r are positive and are in A.P., the roots of...

If $p, q, r$ are positive and are in A.P., the roots of quadratic equation $p x^{2}+q x+r=0$ are all real for
(A) $\left|\frac{r}{p}-7\right| \geq 4 \sqrt{3}$
(B) $\left|\frac{p}{r}-7\right| \geq 4 \sqrt{3}$
(C) all $p$ and $r$
(D) no $p$ and $r$

Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence in which each term after the first is obtained by adding a constant difference to the preceding term. When coefficients of a polynomial are in arithmetic progression, there exists a defined linear relation between them, which can simplify the analysis of the polynomial, particularly by reducing the number of independent variables.

Quadratic Discriminant

The quadratic discriminant is the expression derived from the standard form of the quadratic equation, typically given as b² - 4ac. It determines the nature of the roots of the equation: if the discriminant is non-negative, the quadratic equation has real roots, while a negative discriminant indicates that the roots are complex.

Coefficient Relationship from AP

When the coefficients of a quadratic equation are in arithmetic progression, they satisfy a specific relationship (for instance, twice the middle term equals the sum of the first and the third). This relationship allows one coefficient to be expressed in terms of the others, which then facilitates the substitution into the discriminant expression, leading to conditions that must be met for the roots to be real.

Inequality Analysis

After substituting the coefficient relationship into the quadratic discriminant, the condition for real roots is derived as an inequality involving the remaining coefficients or their ratios. Solving this inequality provides necessary and sufficient conditions (typically formulated as an absolute value inequality) that must be satisfied to ensure that the quadratic equation has all real roots.

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