Algebra 1 / Unit 9: Quadratic Functions and Equations
9.6 The Quadratic Formula
Notes
The quadratic formula solves any $ax^2 + bx + c = 0$:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
The discriminant $b^2 - 4ac$ tells how many real solutions there are:
- Positive: two real solutions.
- Zero: one real solution (a double root).
- Negative: no real solutions.
Practice
For $2x^2 - 3x - 5 = 0$, what is the discriminant?
$(-3)^2 - 4(2)(-5) = 9 + 40$.
Solve $x^2 + 2x - 4 = 0$.
$\displaystyle x = \frac{-2 \pm \sqrt{4 + 16}}{2} = \frac{-2 \pm 2\sqrt{5}}{2}$.
How many real solutions does $x^2 + 6x + 9 = 0$ have?
$36 - 36 = 0$.
Ready for more? Take the Unit 9 test.