The discriminant is a crucial concept in understanding the nature of solutions for quadratic equations. It is defined as the expression under the square root in the quadratic formula, represented mathematically as \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation in the standard form \(ax^2 + bx + c = 0\).

The value of the discriminant provides insight into the number and type of solutions the quadratic equation possesses:

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, often referred to as a repeated or double root.
• If \(D < 0\), the equation has no real solutions, but instead has two complex (imaginary) solutions.

To illustrate how to use the discriminant, consider the following examples:

1. For the equation \(2x^2 + 3x - 2 = 0\), we identify \(a = 2\), \(b = 3\), and \(c = -2\). Plugging these values into the discriminant formula gives:

\[D = 3^2 - 4 \cdot 2 \cdot (-2) = 9 + 16 = 25\]

Since \(D = 25\) (positive), this equation has two real solutions.

2. For the equation \(4x^2 + x + 2 = 0\), we have \(a = 4\), \(b = 1\), and \(c = 2\). The discriminant calculation is:

\[D = 1^2 - 4 \cdot 4 \cdot 2 = 1 - 32 = -31\]

Here, \(D = -31\) (negative), indicating that there are no real solutions, only two imaginary solutions.

3. Lastly, for the equation \(x^2 - 10x + 25 = 0\), we find \(a = 1\), \(b = -10\), and \(c = 25\). The discriminant is calculated as follows:

\[D = (-10)^2 - 4 \cdot 1 \cdot 25 = 100 - 100 = 0\]

With \(D = 0\), this equation has one real solution.

Understanding the discriminant allows for quick assessments of quadratic equations without the need for complete solutions, making it a valuable tool in algebra.