The Discriminant
The discriminant b² - 4ac from the quadratic formula determines how many real solutions a quadratic equation has — a decisive classification tool in enVision Algebra 1 Chapter 9 for Grade 11. If b² - 4ac > 0, the equation has 2 distinct real solutions. If b² - 4ac = 0, it has exactly 1 real solution (a repeated root). If b² - 4ac < 0, it has no real solutions. For 3x² + 5x - 8 = 0: discriminant = 25 + 96 = 121 > 0, so 2 real solutions. For 4y² - 28y + 49 = 0: discriminant = 784 - 784 = 0, so exactly 1 real solution.
Key Concepts
In the Quadratic Formula, $x = \dfrac{ b \pm \sqrt{b^2 4ac}}{2a}$, the quantity $b^2 4ac$ is called the discriminant. For a quadratic equation of the form $ax^2 + bx + c = 0$, $a \neq 0$: If $b^2 4ac 0$, the equation has 2 real solutions. If $b^2 4ac = 0$, the equation has 1 real solution. If $b^2 4ac < 0$, the equation has no real solutions.
Common Questions
What is the discriminant and how do you compute it?
The discriminant is b² - 4ac from the quadratic equation ax² + bx + c = 0. Identify a, b, and c, then compute b² - 4ac.
For 3x² + 5x - 8 = 0, what does the discriminant tell you?
a=3, b=5, c=-8. Discriminant = 25 - 4(3)(-8) = 25 + 96 = 121 > 0. There are 2 distinct real solutions.
For 4y² - 28y + 49 = 0, how many solutions are there?
a=4, b=-28, c=49. Discriminant = 784 - 784 = 0. There is exactly 1 real solution (a double root).
What does a negative discriminant mean geometrically?
A negative discriminant means the parabola does not intersect the x-axis. The quadratic has no real zeros, and the graph sits entirely above or below the x-axis.
Can you use the discriminant without solving the full quadratic formula?
Yes. Computing just b² - 4ac tells you the number of solutions without finding them. This saves time when you only need to know how many solutions exist.