If the absolute value of the difference of roots of the equation x^2+p x+1=0 exceeds, √(3 p), th

step 1 I have given the equation x square plus px plus 1 equal to 0, the absolute difference of the two

If the absolute value of the difference of roots of the...

Key Concepts

Inequalities Involving Parameters

Solving inequalities that involve parameters, especially where the inequality includes square root expressions, requires careful consideration of domain restrictions and sign analysis. It involves manipulating the inequality, squaring both sides (if necessary), and analyzing the resulting conditions to determine the parameter values that satisfy the original inequality.

Difference of Roots

The difference between the two roots of a quadratic equation can be expressed in terms of the discriminant. Specifically, the absolute difference is equal to the square root of the discriminant divided by the absolute value of the leading coefficient. This concept is essential when comparing the numerical separation of the roots to another quantity, such as sqrt(3p).

Quadratic Equations and Their Roots

A quadratic equation is typically written in the form ax² + bx + c = 0, and its solutions (roots) can be calculated using the quadratic formula. Understanding the general behavior of quadratic equations lays the foundation for analyzing relationships between the coefficients (such as p in this example) and the nature of the roots.

Discriminant

The discriminant of a quadratic equation, given by b² - 4ac, determines the nature and number of the roots. It is also central in expressing the difference between the roots, as this difference is calculated using the square root of the discriminant. A positive discriminant indicates two distinct real roots, while a negative discriminant leads to complex roots.

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