Topic 7: Quadratics

Core and Extension

Distributive Laws

a(b±c) = ab±ac

(a+b)(c+d) = ac+ad+bc+bd

Perfect Squares

(a+b)² = a²+2ab+b²

(a-b)² = a²-2ab+b²

Difference of Perfect Squares (DOPS)

(a+b)(a-b) = a²-b²

Factorisation

1. Taking out the common factors

2. Grouping terms (X = ac, + = b)

   e.g. x²+3x-10 → 5*-2 = -10 = ac, 5+(-2) = c = b

   = x²+5x-2x-10

   = x(x+5)-2(x+5)

   = (x-2)(x+5)

Monic Quadratics

x²+(m+n)x+mn = (x+m)(x+n)

Non - Monic Quadratic Trinomials

1. Two numbers to give a*c and b when a+c;

2. Split bx, then factorise by grouping.

e.g. 6x²+19x+60

15×4 = 60

∴ 6x²+4x+15x+60

=3x(2x+5) + 2(2x+5)

= (3x+2)(2x+5)

Tip: How to find perfect squares (completely optionaal ;)

1. Check if a/c are square numbers

2. check if a*c/2 = b

3. If all are satisfied, then it is a perfect square (depending on the positivity of bx; if positive, the square is positive, or vice versa).

e.g. 16x²-40x+25

16x² = (4x)²

25 = (5)²

16×5/2 = 40 (ignore pos/neg here)

∴(4x-5)².

Completing the Square

BHS Rule: add and substract (b/2)² - factorise by DOPS (surd allowed).

Quadratics Formula and the Discriminant

Quadratics Formula: (-b±√(b²-4ac))/(2a).

Discriminant (denoted by the greek letter delta ‘Δ’): b²-4ac

When Δ < 0, no real solutions (as √Δ = √-num, undefined)

When Δ = 0, real solution (x = -b/2a)

When Δ > 0, 2 real solutions (x = quadratics formula)

Turning Point

Formula: a(x±h)²+k=0, (±h, k) as turning point.

Second Formula: h = -b/2a, k = c-b²/4a

Axis of symmetry: x = h/-h

Tip: y-int = c