Roots of a Quadratic Equation

To determine the number of real roots of a quadratic equation, we use the discriminant. A quadratic equation is typically written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are real numbers and \(a \neq 0\).

Understanding the Discriminant

The discriminant, denoted by the symbol \(\Delta\) (or \(D\)), is a crucial part of the quadratic formula. It helps us understand the nature of the roots without actually solving the equation. The formula for the discriminant is:

\[ \Delta = b^2 - 4ac \]

Based on the value of the discriminant, we can determine the number and type of roots:

• If \(\Delta > 0\), the quadratic equation has two distinct real roots.
• If \(\Delta = 0\), the quadratic equation has exactly one real root (also called a repeated or double root).
• If \(\Delta < 0\), the quadratic equation has no real roots. Instead, it has two distinct complex conjugate roots.

Calculating the Discriminant for \(3x^2 + 4x + 25 = 0\)

Let's identify the coefficients \(a\), \(b\), and \(c\) from the given quadratic equation: \(3x^2 + 4x + 25 = 0\).

• Coefficient \(a = 3\)
• Coefficient \(b = 4\)
• Constant term \(c = 25\)

Now, we will substitute these values into the discriminant formula:

\[ \Delta = b^2 - 4ac \]

\[ \Delta = (4)^2 - 4(3)(25) \]

\[ \Delta = 16 - 12(25) \]

\[ \Delta = 16 - 300 \]

\[ \Delta = -284 \]

Interpreting the Result of the Discriminant

Our calculated discriminant is \(\Delta = -284\).

Since \(\Delta = -284\), which is less than zero (\(\Delta < 0\)), the quadratic equation \(3x^2 + 4x + 25 = 0\) has no real roots.

This means there are "nil" real roots for the given quadratic equation. The roots, in this case, would be complex numbers.