Algebra › Quadratics
Quadratics
Quadratics are the equations where x is squared. Their graphs are parabolas — the U-shaped curves you see in fountains and ball trajectories.
y = ax² + bx + c
y = 1x² + 0x − 2
Vertex: (0.00, -2.00)
Discriminant b² − 4ac = 8.0 (two roots)
The standard form
A quadratic looks like y = ax² + bx + c, where a ≠ 0. The values of a, b, c shape the curve:
- a > 0 — opens upward (smile).
- a < 0 — opens downward (frown).
- Bigger |a| — narrower curve.
- c — y-intercept (where the curve hits the y-axis).
Vertex (the tip)
The lowest (or highest) point of the parabola is the vertex. Its x-coordinate is x = −b / (2a). Plug that back into the equation to get the y-coordinate.
Roots — where y = 0
The roots are the x-values where the curve crosses the x-axis. The quadratic formula gives them all in one go:
x = (−b ± √(b² − 4ac)) / (2a)
The discriminant
b² − 4ac, decides how many real roots there are. Positive → two roots. Zero → one. Negative → none (the parabola never touches the x-axis).Where you'll see them
- The path of a thrown ball.
- The shape of a satellite dish or car headlight reflector.
- Profit/cost curves in business problems.
If b² − 4ac = 0, how many real roots does the quadratic have?
For y = x² − 4x + 1, what is the x-coordinate of the vertex?