Integrated Math 2 / Unit 4: Solving Quadratic Equations
4.5 The Discriminant
Notes
The discriminant $b^2 - 4ac$ tells how many real solutions a quadratic has.
- Positive: two real solutions (the parabola crosses the $x$-axis twice).
- Zero: one real solution (the vertex touches the $x$-axis).
- Negative: no real solutions (two complex solutions).
- Example: $x^2 + 2x + 5$ has $4 - 20 = -16 < 0$, so no real solutions.
Practice
How many real solutions does $x^2 + 4x + 4 = 0$ have?
The discriminant is $16 - 16 = 0$.
How many real solutions does $x^2 + 2x + 5 = 0$ have?
The discriminant is $4 - 20 = -16$, which is negative.
For which value of $k$ does $x^2 + 6x + k = 0$ have exactly one real solution?
Set $36 - 4k = 0$, so $k = 9$.
Ready for more? Take the Unit 4 test.