Algebra 2 / Unit 2: Quadratic Functions and Complex Numbers
2.5 The Discriminant
Notes
The discriminant is $b^2 - 4ac$, the part under the square root in the quadratic formula. It tells you the number and type of solutions.
- $b^2 - 4ac > 0$: two real solutions (graph crosses the $x$-axis twice).
- $b^2 - 4ac = 0$: one real solution, a double root (vertex touches the $x$-axis).
- $b^2 - 4ac < 0$: two nonreal complex solutions (graph misses the $x$-axis).
- Example: $x^2 + 2x + 5 = 0$ has $4 - 20 = -16 < 0$, so two complex solutions.
Practice
How many real solutions does $x^2 - 6x + 9 = 0$ have?
$(-6)^2 - 4(1)(9) = 0$, so there is one double root.
Describe the solutions of $2x^2 + x + 5 = 0$.
$1 - 4(2)(5) = -39 < 0$.
For which values of $k$ does $x^2 + kx + 16 = 0$ have exactly one real solution?
Set $k^2 - 64 = 0$, so $k = 8$ or $k = -8$.
Ready for more? Take the Unit 2 test.