Quadratic inequalities and parameters
Solve quadratic inequalities by the signs of their factors, and see how a letter in an equation turns 'how many roots?' into an inequality.
- UK: Year 10
- Iran: grade 10
- Afghanistan: grade 9
- Syria: grade 10
- Palestine: grade 10
- Ukraine: grade 9
- Russia: grade 9
Guide
This workshop follows 'signs of products'. Here a quadratic such as x2 − 4 is first split into factors: (x − 2)(x + 2).
A product of two factors is negative when the factors have opposite signs. The zeros of the factors cut the line into gaps; in each gap write the sign of each factor, and get the sign of the product from the rule of signs.
A square is never negative: (x − 1)2 ≥ 0. So in (x − 1)2 > 0 only the one point x = 1 fails.
Sometimes an equation has another letter in it, such as a in x2 + ax + 1 = 0. The discriminant Δ = a2 − 4 tells you how many roots there are. Two roots means Δ > 0, which is itself a quadratic inequality, but now in the letter a.
Tap the line to get a piece, choose its shape with the buttons, and tap a dot to make it filled or hollow. Under the line the answer is written as an inequality and in interval notation.