Using b² − 4ac to describe the roots, solving equations that turn into quadratics, and using α + β and αβ without solving.
5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
You can tell how many roots a quadratic has — and find expressions in the roots — without ever solving it.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
See how the discriminant decides the number of roots.
Copy these into your notebook in your own words — that is how they stick.
D = b² − 4ac: > 0 two real roots; = 0 equal roots; < 0 no real roots.
D ≥ 0.
A line touches a curve when the combined quadratic has D = 0.
u = x², u = √x, u = 2ˣ …; solve for u, then back to x; reject impossible u.
α + β = −b/a; αβ = c/a.
α² + β² = (α + β)² − 2αβ; 1/α + 1/β = (α + β)/(αβ); (α − β)² = (α + β)² − 4αβ.
x² − (sum of new roots)x + (product of new roots) = 0.
Substitute a root or compare with the original sum and product.
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Additional Mathematics Arcade.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (26 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [14 marks]
The roots of the equation 3x² − 9x + 1 = 0 are α and β.
(a) Write down α + β and αβ. [2]
(b) Without solving the equation, find the value of (i) α² + β², (ii) 1/α + 1/β. [5]
(c) Find an equation, with integer coefficients, whose roots are 1/α and 1/β. [4]
(d) Show that the equation 3x² − 9x + k = 0 has real roots only if k ≤ 27/4. [3]
Question 2 [12 marks]
Answer all parts.
(a) Solve x⁴ − 13x² + 36 = 0. [4]
(b) Solve x − 5√x + 6 = 0. [4]
(c) Show that the line y = x + c is a tangent to the circle x² + y² = 8 when c = ±4. [4]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.