Expanding and Factorising Quadratics — Solutions

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1. Expand using FOIL

   1. (a) (x + 4)(x + 2): F: x², O: 2x, I: 4x, L: 8.   x² + 6x + 8
   2. (b) (x − 5)(x + 3): F: x², O: 3x, I: −5x, L: −15.   x² − 2x − 15
   3. (c) (2x + 1)(x + 4): F: 2x², O: 8x, I: x, L: 4.   2x² + 9x + 4
   4. (d) (3x − 2)(2x + 5): F: 6x², O: 15x, I: −4x, L: −10.   6x² + 11x − 10
   5. (e) (x + 2y)(x − 3y): F: x², O: −3xy, I: 2xy, L: −6y².   x² − xy − 6y²
   6. (f) (4 − x)(3 + 2x): F: 12, O: 8x, I: −3x, L: −2x².   12 + 5x − 2x²

2. Special products

   1. (a) (x + 6)²: (a+b)² = a²+2ab+b².   x² + 12x + 36
   2. (b) (2x − 3)²: (a−b)² = a²−2ab+b²: (2x)²−2(2x)(3)+9 = 4x² − 12x + 9
   3. (c) (x + 8)(x − 8): Difference of two squares: a²−b² = x² − 64
   4. (d) (5x + 2)(5x − 2): Difference of two squares: (5x)²−2² = 25x² − 4
   5. (e) (3x + y)²: (3x)²+2(3x)(y)+y² = 9x² + 6xy + y²
   6. (f) (1 − 4x)²: 1²−2(1)(4x)+(4x)² = 1 − 8x + 16x²

3. Factorise monic quadratics

   1. (a) x² + 7x + 12: Find p×q=12, p+q=7: p=3, q=4.   (x + 3)(x + 4)
   2. (b) x² − 8x + 15: p×q=15, p+q=−8: p=−3, q=−5.   (x − 3)(x − 5)
   3. (c) x² + x − 20: p×q=−20, p+q=1: p=5, q=−4.   (x + 5)(x − 4)
   4. (d) x² − 2x − 35: p×q=−35, p+q=−2: p=−7, q=5.   (x − 7)(x + 5)
   5. (e) x² − 10x + 25: p×q=25, p+q=−10: p=q=−5. Perfect square.   (x − 5)²
   6. (f) x² − 49: Difference of two squares: x²−7².   (x − 7)(x + 7)

4. HCF then factorise

   1. (a) 3x² + 9x: HCF = 3x.   3x(x + 3)
   2. (b) 2x² − 18: HCF = 2. → 2(x²−9) = 2(x−3)(x+3).   2(x − 3)(x + 3)
   3. (c) 4x² − 8x − 12: HCF = 4. → 4(x²−2x−3) = 4(x−3)(x+1).   4(x − 3)(x + 1)
   4. (d) 5x² − 20x + 20: HCF = 5. → 5(x²−4x+4) = 5(x−2)².   5(x − 2)²

5. Non-monic quadratics

   1. (a) 2x² + 5x + 2: ac=4, sum=5: 1×4. Split: 2x²+x+4x+2 = x(2x+1)+2(2x+1).   (2x + 1)(x + 2)
   2. (b) 3x² − 7x + 2: ac=6, sum=−7: −1×−6. Split: 3x²−x−6x+2 = x(3x−1)−2(3x−1).   (3x − 1)(x − 2)
   3. (c) 4x² + 12x + 9: ac=36, sum=12: 6×6. Split: 4x²+6x+6x+9 = 2x(2x+3)+3(2x+3).   (2x + 3)²
   4. (d) 6x² + x − 2: ac=−12, sum=1: 4×−3. Split: 6x²+4x−3x−2 = 2x(3x+2)−1(3x+2).   (2x − 1)(3x + 2)
   5. (e) 2x² − x − 10: ac=−20, sum=−1: 4×−5. Split: 2x²+4x−5x−10 = 2x(x+2)−5(x+2).   (2x − 5)(x + 2)

6. Difference of squares and complete factorisation

   1. (a) 4x² − 25: (2x)² − 5².   (2x − 5)(2x + 5)
   2. (b) 9x² − y²: (3x)² − y².   (3x − y)(3x + y)
   3. (c) 16a² − 81b²: (4a)² − (9b)².   (4a − 9b)(4a + 9b)
   4. (d) x⁴ − 81: (x²)²−9² = (x²−9)(x²+9) = (x−3)(x+3)(x²+9).   (x − 3)(x + 3)(x² + 9)
   5. (e) 3x² − 75: HCF=3: 3(x²−25) = 3(x−5)(x+5).   3(x − 5)(x + 5)

7. Rectangle dimensions

   1. (a) Area: A = (x+5)(x+2) = x²+2x+5x+10.   A = x² + 7x + 10
   2. (b) Solve: x²+7x+10=40 ⇒ x²+7x−30=0 ⇒ (x+10)(x−3)=0 ⇒ x=3 (since x>0).   x = 3
   3. (c) Dimensions and perimeter: Length = 3+5 = 8 m, Width = 3+2 = 5 m. P = 2(8+5) = 26 m

8. Special products in context

   1. (a) 103 × 97: = (100+3)(100−3) = 100²−3² = 10000−9.   9991
   2. (b) (x+5)²+(x−5)²: = (x²+10x+25)+(x²−10x+25) = 2x²+50. The x-terms cancel — the sum contains no linear term.   2x² + 50
   3. (c) 4x²−20x+25: = (2x)²−2(2x)(5)+5² = (2x−5)² — a perfect square trinomial.
   4. (d) 3x²−3: HCF=3: 3(x²−1) = 3(x−1)(x+1).   3(x − 1)(x + 1)

9. Multi-step simplification

   1. (a) (x+2)²−(x−4)(x+3): = (x²+4x+4)−(x²−x−12) = x²+4x+4−x²+x+12 = 5x + 16
   2. (b) 5x+16=36: 5x=20.   x = 4
   3. (c) Always positive for x>−3.2: 5x+16 = 0 when x = −16/5 = −3.2. For x>−3.2: 5x>−16, so 5x+16>0. The expression is always positive for x > −3.2.

10. Algebraic proof

    1. (a) Prove (a+b)²−(a−b)²=4ab: LHS = (a²+2ab+b²)−(a²−2ab+b²) = 4ab = RHS.   Proven. ✓
    2. (b) 75²−25²: Using a=75, b=25: (75+25)(75−25) = 100×50 = 5000
    3. (c) Consecutive even integers: Let integers be 2n and 2n+2. (2n+2)²−(2n)² = [(2n+2)−2n][(2n+2)+2n] = 2×(4n+2) = 4(2n+1). Since 4 is a factor, the difference is always divisible by 4. ✓