Quadratic equation with a parameter
Slide a and watch the parabola go up and down: the number of roots follows the discriminant, the signs of the roots follow Vieta's theorem.
- Russia: grade 8
Guide
This is an advanced topic: in the Russian programme it is in the advanced level of Grade 8 ("quadratic equation with a parameter"). It can be skipped in an ordinary lesson.
A parameter is a letter that stands for a fixed but unspecified number. In x2 − 2x + a = 0 the parameter a changes the number of roots. Here D = 4 − 4a: two roots when D > 0, one when D = 0, none when D < 0.
In the picture the parabola y = x2 − 2x + a moves up and down with the slider. The roots are the points where it meets the x-axis. For the signs of the roots use Vieta: the product of the roots is the constant term and their sum is minus the coefficient of x.
Slide the slider in the first tasks and count the crossings. In the others type the answer or choose an option.