91Ó°ÊÓ

A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a, b\), and \(c\) are constants and \(a eq 0\). Quadratic equations are fundamental in algebra and appear in many fields of science. The general solution to a quadratic equation is given by the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. This formula helps us find the values of \(x\) where the quadratic equation equals zero, known as the roots or solutions of the equation.

In our exercise, the quadratic equation \(4x^2 + 7x - 2\) is found inside the square root function. To find where this function is defined, we need to solve for the values of \(x\) such that the expression under the square root is non-negative (i.e., \(4x^2 + 7x - 2 \geq 0\)). This is key for determining the domain of the function.

Interval notation is a way of writing subsets of the real number line. It uses parentheses and brackets to describe the set of numbers between two endpoints. Parentheses \(()\) denote open intervals where the endpoints are not included, while brackets \([]\) indicate closed intervals where the endpoints are included.

For example, \((-2, 5)\) represents all numbers between \(-2\) and \(5\), not including \(-2\) and \(5\) themselves. Conversely, \([-2, 5]\) includes both \(-2\) and \(5\).

In the context of our problem, once we find the roots of the quadratic equation, we use interval notation to describe the domain of the function. Here we determined that \(4x^2 + 7x - 2\) is non-negative in the intervals \((-\infty, -2]\) and \([\frac{1}{4}, +\infty)\). These intervals describe where the function \(q(x)=\sqrt{4x^2+7x-2}\) is defined and hence forms its domain.

The discriminant is a part of the quadratic formula and is crucial in understanding the nature of the roots of a quadratic equation. It is given by the expression \(b^2 - 4ac\).

The discriminant can tell us the following:

• If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.


• If \(b^2 - 4ac = 0\), the quadratic equation has exactly one real root (also called a double root).


• If \(b^2 - 4ac < 0\), the quadratic equation has no real roots but two complex roots.

For our quadratic equation \(4x^2 + 7x - 2\), we calculated the discriminant as follows:
\(b^2 - 4ac = 7^2 - 4(4)(-2) = 49 + 32 = 81\). Since the discriminant is positive (\(81 > 0\)), we know that the quadratic equation has two distinct real roots, \(-2\) and \(\frac{1}{4}\). These roots are critical in determining the intervals of our domain.