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

If $p, q, r$ are positive and are in A.P, the roots of the 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) $\left|\frac{q}{p}-4\right| \geq 2 \sqrt{3}$ (c) $\left|\frac{p}{r}-7\right| \geq 4 \sqrt{3}$ (d) $\left|\frac{p}{q}-1\right| \geq \frac{\sqrt{3}}{2}$

If $p, q, r$ are positive and are in A.P, the roots of the 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) $\left|\frac{q}{p}-4\right| \geq 2 \sqrt{3}$
(c) $\left|\frac{p}{r}-7\right| \geq 4 \sqrt{3}$
(d) $\left|\frac{p}{q}-1\right| \geq \frac{\sqrt{3}}{2}$

Key Concepts

Quadratic Discriminant

For a quadratic equation ax² + bx + c = 0, the nature of its roots is determined by the discriminant (b² - 4ac). The roots are real if and only if the discriminant is greater than or equal to zero. This concept is essential when analyzing any conditions that ensure the quadratic has real solutions.

Arithmetic Progression

An arithmetic progression is a sequence in which each term after the first is obtained by adding a constant difference. When applied to the coefficients of a quadratic equation, this condition establishes a specific relationship (typically 2q = p + r) among the coefficients, which can be used to express one coefficient in terms of the others.

Coefficient Relationships in Quadratics

When the coefficients of a quadratic are related by additional conditions such as being in an arithmetic progression, these relationships reduce the number of independent variables. This simplification can be exploited to express the discriminant in terms of a single variable or coefficient ratio, thereby easing the derivation of conditions (often inequalities) that ensure real roots.

Inequalities and Absolute Value Analysis

Often, conditions for real roots in modified quadratic equations lead to inequalities involving ratios of coefficients. The use of absolute values in these inequalities accounts for the magnitude of difference required between coefficients, ensuring that the formulated conditions cover all valid cases for the quadratic to have real roots.

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