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RV-01-001 · Double-bracket factorising: paired monic quadratics

Factorise each expression into two brackets.

(a) x2+22x+96x^{2} + 22x + 96

(b) x2+4x−60x^{2} + 4x - 60

Part (a) Factorising: Quadratic Expressions a = 1 (Grid Method) Booklet p. 5
Method
1) a × c
2) List factors of ac
3) Select pair which add/subtract to give b
4) Fill in 2×2 grid
5) Factorise outside grid
6) Write brackets
Worked Example (RV-01-001 · Part (a))
x2+22x+96x^{2} + 22x + 96
Factorising Grid (Booklet p. 5 — Circle HCF xx on left):
×xx+6+6
xxx2x^2+6x+6x
+16+16+16x+16x+96+96
a×c=1×(96)=96a \times c = 1 \times (96) = 96
+1+1+96+96
+2+2+48+48
+3+3+32+32
+4+4+24+24
+6+6+16+16
+8+8+12+12
c is positive so both signs are the same (+ and +)...
+6+16=+22+6 + 16 = +22 ✔
Brackets: (x+16)(x+6)\text{Brackets: } \mathbf{(x + 16)(x + 6)}
(x+16)(x+6)(x + 16)(x + 6)
Part (b) Factorising: Quadratic Expressions a = 1 (Grid Method) Booklet p. 5
Method
1) a × c
2) List factors of ac
3) Select pair which add/subtract to give b
4) Fill in 2×2 grid
5) Factorise outside grid
6) Write brackets
Worked Example (RV-01-001 · Part (b))
x2+4x−60x^{2} + 4x - 60
Factorising Grid (Booklet p. 5 — Circle HCF xx on left):
×xx−6-6
xxx2x^2−6x-6x
+10+10+10x+10x−60-60
a×c=1×(−60)=−60a \times c = 1 \times (-60) = -60
−1-1+60+60
−2-2+30+30
−3-3+20+20
−4-4+15+15
−5-5+12+12
−6-6+10+10
c is negative so signs are different (+ and −)...
−6+10=+4-6 + 10 = +4 ✔
Brackets: (x+10)(x−6)\text{Brackets: } \mathbf{(x + 10)(x - 6)}
(x+10)(x−6)(x + 10)(x - 6)