quadratic equation

Tier 1 (Memorisation) and Tier 2 (Procedural) knowledge

ax² + bx + c = 0

What is the standard form of a quadratic equation?

x = [-b ± √(b²-4ac)] / 2a

What is the Quadratic Formula for solving ax² + bx + c = 0?

Δ = b² - 4ac

What is the formula for the discriminant?

Δ > 0 → Two distinct real roots
Δ = 0 → One repeated real root
Δ < 0 → No real roots

What are the three discriminant rules for the nature of roots?

y = a(x - h)² + k

What is the vertex form of a quadratic function?

(h, k)

From y = a(x - h)² + k, what are the coordinates of the vertex?

x = h

From y = a(x - h)² + k, what is the equation of the axis of symmetry?

a > 0 → Minimum value
a < 0 → Maximum value

How does the sign of 'a' determine if the vertex is a maximum or minimum?

Procedure: Solving by Factorisation
1. Standard Form: 2x² - 5x - 3 = 0
2. Factorise: (2x + 1)(x - 3) = 0
3. Set factors to zero: 2x + 1 = 0 or x - 3 = 0
4. Solve: x = -1/2 or x = 3

Problem: Solve 2x² - 5x - 3 = 0 by factorisation.

Procedure: Solving using Quadratic Formula
1. Standard Form: 3x² - 7x - 5 = 0
2. Identify a=3, b=-7, c=-5
3. Substitute into formula: x = [-(-7) ± √((-7)² - 4(3)(-5))] / 2(3)
4. Calculate: x = [7 ± √(49 + 60)] / 6 = [7 ± √109] / 6

Problem: Solve 3x² - 7x - 5 = 0 using the formula.

Procedure: Completing the Square (a≠1)
1. Factor out 'a': -3[x² - 6x] - 10
2. Complete square inside: -3[(x - 3)² - 3²] - 10
3. Expand & Simplify: -3(x - 3)² + 27 - 10
4. Final Form: -3(x - 3)² + 17

Problem: Express -3x² + 18x - 10 in vertex form.

Procedure: Graph Sketching
1. Orientation: a=1 > 0, so U-shape.
2. Y-intercept (x=0): y = -8. Point (0, -8).
3. X-intercepts (y=0): (x-4)(x+2)=0. Points (4, 0) and (-2, 0).
4. Turning Point: y = (x-1)² - 9. Vertex (1, -9).
5. Plot points and draw a smooth curve.

Problem: What are the key steps to sketch the graph of y = x² - 2x - 8?

Procedure: Graphical Solution with a Line
1. Target Eq: 2x² - 3x - 4 = 0
2. Isolate Original Function: (2x² - x) - 2x - 4 = 0
3. Identify the Line: 2x² - x = 2x + 4
4. Conclusion: Draw the line y = 2x + 4

Problem: The graph of y = 2x² - x is drawn. What line must be drawn to solve 2x² - 3x - 4 = 0?