91Ó°ÊÓ

A quadratic equation is a type of polynomial equation that has the standard form: \[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are coefficients, and \( a eq 0 \) because if \( a \) were 0, the equation would be linear, not quadratic. Quadratic equations are essential in algebra and are characterized by their highest exponent, which is a power of 2. This means that their graphs are parabolas, either opening upwards or downwards, depending on the sign of \( a \).
Common features of quadratic equations include:

• The leading coefficient \( a \) determines the "width" and "direction" of the parabola.
• The constant term \( c \) affects the vertical position of the parabola on a graph.

Solving these equations usually involves finding the points (or roots) where the graph intersects the x-axis, i.e., where \( y = 0 \). The quadratic formula is a systematic way to find these roots.

The discriminant is a vital part of the quadratic formula and provides important information about the nature of the roots of a quadratic equation. It is derived from the formula \[ b^2 - 4ac \]and directly influences the number and type of roots:

• If the discriminant is positive (\( b^2 - 4ac > 0 \)), the quadratic equation has two distinct real roots.
• If it is zero (\( b^2 - 4ac = 0 \)), there is exactly one real root, indicating a perfect square scenario where the graph touches the x-axis at one point.
• If negative (\( b^2 - 4ac < 0 \)), the equation has no real roots, but two complex imaginary roots.

In our example, the discriminant was calculated to be 24, which is positive. This suggests that the quadratic equation has two distinct and real roots. Knowing the discriminant helps predict the behavior of the parabolic curve before solving the equation fully.

The roots of a quadratic equation are the solutions to the equation, or the values of \( x \) where the equation equals zero. These roots correspond to the x-values where the parabola of the quadratic equation intersects the x-axis. To find these roots, we use the quadratic formula, \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \].Let's break down this process:

• Substitute the coefficients \( a \), \( b \), and \( c \) into the formula.
• Calculate the discriminant \( b^2 - 4ac \) to determine the nature of the roots.
• Plug the discriminant into the formula to find the values of \( x \).

For our specific case, using the coefficients \( a = 1 \), \( b = -2 \), and \( c = -5 \), we found the roots to be \( x = 1 + \sqrt{6} \) and \( x = 1 - \sqrt{6} \). These are real and distinct roots, confirming what we anticipated from the positive discriminant. Understanding the roots is essential for graphing quadratic functions and solving real-world problems where these models apply.