Quadratic equations
Standard form | [[$ ax^2+bx+c=0, \quad a\neq 1 $]] |
Quadratic formula: | [[$ \quad x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a} $]] |
Parabola opening direction and shape:
- If [[$ a > 0 $]], the parabola opens upwards.
- If [[$ a < 0 $]], the parabola opens downwards.
- If [[$ |a| $]] is small, the parabola is wide.
- If [[$ |a| $]] is great, the parabola is narrow.
Incomplete quadratic equations
The number of solutions to equation [[$ ax^2 + c = 0 $]] depends on the constant [[$c$]]:
- [[$c < 0$]]: two solutions, solutions of opposite numbers
- [[$c= 0$]]: the only solution is [[$x = 0$]]
- [[$c > 0$]]: no solution
- always two solutions, the other is always [[$x = 0$]].