91Ó°ÊÓ

The quadratic formula is a powerful tool used to solve any quadratic equation, which typically takes the form \( ax^2 + bx + c = 0 \). This formula is an essential mathematical method because it provides a way to find the roots (solutions) of the equation regardless of whether they are real or complex. The quadratic formula is given as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation, where \( a eq 0 \).
- **Step by Step Usage:** - Identify values of \( a \), \( b \), and \( c \) from the equation. - Substitute these values into the quadratic formula. - Carry out the arithmetic operations to simplify the expression and find the values of \( x \).
Using the quadratic formula is efficient, especially when factoring is complex or impossible. This is why it is preferred over other methods in many algebra scenarios.

The discriminant is a part of the quadratic formula, represented by \( b^2 - 4ac \). It plays a crucial role in determining the nature of the roots of a quadratic equation. By calculating the discriminant, one can predict whether the roots will be real or complex, and if they are real, whether they will be distinct or repeated.
- **Interpretation of the Discriminant:** - If \( b^2 - 4ac = 0 \), the equation has one real double root. This means the parabola touches the x-axis at a single point. - If \( b^2 - 4ac > 0 \), the equation has two distinct real roots. The parabola intersects the x-axis at two separate points, as seen in our exercise. - If \( b^2 - 4ac < 0 \), the equation has two complex roots that are conjugates of each other. This implies the parabola does not touch the x-axis at all.
Calculating the discriminant is often the first step when using the quadratic formula as it gives a quick insight into the possible solutions.

Real roots are solutions to the quadratic equation that are actual numbers (not imaginary or complex). When a quadratic equation's discriminant is positive, you will always find two distinct real roots.
- **Characteristics of Real Roots:** - They are the points where the parabola associated with the quadratic equation crosses the x-axis. - If they are rational, they can often be expressed as fractions. - If irrational, the roots are non-repeating, non-terminating decimals.
In our exercise example, after applying the quadratic formula and calculating the discriminant, we found two real roots for the equation \( 6n^2 - 31n + 40 = 0 \), which were \( n = \frac{8}{3} \) and \( n = \frac{5}{2} \). These values can be tested by plugging them back into the original equation, ensuring they satisfy the equation, thus confirming them as real solutions.