Understanding the Discriminant

Today, we're diving into the final method for analyzing quadratic functions: the discriminant. The discriminant is a powerful tool that tells us how many solutions a quadratic equation has without actually solving for them. This is super useful, especially when you just need to know the type of solutions you're dealing with.

Before we get started, remember that a quadratic equation needs to be in standard form: $ax^2 + bx + c = 0$. Once you have it in this form, identifying $a$, $b$, and $c$ is easy!

What is the Discriminant?

The discriminant is the part of the quadratic formula under the square root: $b^2 - 4ac$. We use this value to determine the nature of the roots (solutions) of the quadratic equation.

  • If $b^2 - 4ac > 0$: The equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points.
  • If $b^2 - 4ac = 0$: The equation has exactly one real solution (a repeated root). The parabola touches the x-axis at exactly one point.
  • If $b^2 - 4ac < 0$: The equation has two complex solutions. The parabola does not intersect the x-axis. These solutions involve imaginary numbers.

Example Time!

Let's look at some examples to solidify our understanding. Consider the quadratic equation $2x^2 - 5x + 3 = 0$.

  1. Identify $a$, $b$, and $c$: In this case, $a = 2$, $b = -5$, and $c = 3$.
  2. Calculate the discriminant: $b^2 - 4ac = (-5)^2 - 4(2)(3) = 25 - 24 = 1$.
  3. Analyze the result: Since $1 > 0$, the equation has two distinct real solutions.

Complex Solutions - A Sneak Peek

What happens when the discriminant is negative? That's when we encounter complex solutions. Complex numbers involve the imaginary unit, $i$, where $i = \sqrt{-1}$. When $b^2 - 4ac < 0$, we end up taking the square root of a negative number, leading to complex solutions. For example, if you get a discriminant of -4, the square root would be $2i$.

Class Assignments

Homework: pg. 105 # 18-29 all. Focus on finding the discriminant for each problem and determining the number and type of solutions.

Discussion Question: In some of the notes for this lesson, it talks about when the value of the discriminant is less than zero, it has two complex solutions. Look online for a site that explains what a complex solution is. Only the first of each site will count for credit.