If the difference between the roots of the equation \(\mathrm{x}^{2}+\mathrm{ax}+1=0\) is less than \(\sqrt{5}\) then the set of possible values of a is (a) \((-\overline{3,3})\) (b) \((-3, \infty)\) (c) \((3, \infty)\) (d) \((-\infty,-3)\)

The quadratic formula is a fundamental tool for solving quadratic equations of the form Ax^2 + Bx + C = 0. It explicitly provides the solutions (also known as roots) which makes it invaluable for various mathematical and practical problems. Given by

\[ x = \frac{{-B \pm \sqrt{{B^2 - 4AC}}}}{{2A}} \]
it takes the coefficients of the equation (A, B, and C) and delivers the roots by incorporating the 'plus-minus' symbol (\(\pm\)) to account for two possible solutions. These solutions can be real or complex numbers and provide insights into the behavior of the quadratic function represented by the equation.

Understanding the Discriminant

Within the quadratic formula, the expression under the square root, \(B^2 - 4AC\), is called the discriminant. It determines the nature of the roots—real and distinct, real and equal, or complex. Positive discriminant indicates two distinct real roots, zero discriminant indicates a single repeated real root, and negative discriminant indicates complex roots.

Inequalities are mathematical expressions that describe the relative size or order of two values. They are central to understanding a wide range of mathematical concepts, particularly when dealing with ranges of solutions or conditions. Unlike equations that assert two quantities are equal, inequalities indicate that one quantity is greater than, less than, greater than or equal to, or less than or equal to another. The symbols used are >, <, >=, <=.

Inequalities become especially important when dealing with domains or ranges of functions, maxima and minima in calculus, and constraints in optimization problems. In the context of quadratic equations, inequalities can be used to describe the range of possible values for the coefficients that meet certain criteria about the roots, such as the exercise example concerning the differences between roots.

Manipulating Inequalities

When we square both sides of an inequality or perform other operations, it's essential to be careful. Squaring both sides is only safe when dealing with positive quantities since it preserves the order. If there is uncertainty about the sign, the inequality must be considered in its separate positive and negative cases to ensure the correct solution.

The difference between roots of a quadratic equation provides important information about the spacing and symmetry of the roots on the number line. For the equation Ax^2 + Bx + C = 0, the roots can be labeled as \(\alpha\) and \(\beta\). The formula for finding the difference is simply

\[ |\alpha - \beta| = |\frac{{-B + \sqrt{{B^2 - 4AC}}}}{{2A}} - \frac{{-B - \sqrt{{B^2 - 4AC}}}}{{2A}}| \]
simplifying to \[ |\alpha - \beta| = \frac{{\sqrt{{B^2 - 4AC}}}}{{|A|}} \]
This difference, given by the magnitude of the square root of the discriminant (\(\sqrt{B^2 - 4AC}\)), divided by the absolute value of \(A\), reveals whether the roots are close to each other or far apart. In particular scenarios, like in the provided exercise, this concept helps determine the range of coefficients that would yield roots with specific characteristics.