Step-by-Step Factorisation of Quadratic Expressions
1. x² + 8x + 15
Step 1: Find two numbers whose product is 15 and sum is 8.
Step 2: The numbers are 3 and 5.
Step 3: Factor as (x + 3)(x + 5)
Answer: (x + 3)(x + 5)
2. x² + 9x + 8
Step 1: Find two numbers that multiply to 8 and add to 9.
Step 2: The numbers are 1 and 8.
Step 3: Factor as (x + 1)(x + 8)
Answer: (x + 1)(x + 8)
3. x² + 12x + 36
Step 1: Recognize it as a perfect square trinomial.
Step 2: 6 × 6 = 36 and 6 + 6 = 12
Step 3: Factor as (x + 6)²
Answer: (x + 6)(x + 6) or (x + 6)²
4. 2h² + 5h − 3
Step 1: Multiply a × c = 2 × (−3) = −6
Step 2: Find two numbers that multiply to −6 and add to 5 → 6 and −1
Step 3: Rewrite middle term: 2h² + 6h − h − 3
Step 4: Group and factor: 2h(h + 3) −1(h + 3)
Step 5: Final factor: (2h − 1)(h + 3)
Answer: (2h − 1)(h + 3)
5. 3x² + 4x + 1
Step 1: a × c = 3 × 1 = 3
Step 2: Find two numbers that multiply to 3 and add to 4 → 3 and 1
Step 3: Rewrite middle term: 3x² + 3x + x + 1
Step 4: Group: 3x(x + 1) + 1(x + 1)
Step 5: Factor: (3x + 1)(x + 1)
Answer: (3x + 1)(x + 1)
6. 3s² + s − 10
Step 1: a × c = 3 × (−10) = −30
Step 2: Numbers that multiply to −30 and add to 1 → 6 and −5
Step 3: Rewrite: 3s² + 6s − 5s − 10
Step 4: Group: 3s(s + 2) −5(s + 2)
Step 5: Factor: (3s − 5)(s + 2)
Answer: (3s − 5)(s + 2)
7. x² − 2x − 15
Step 1: Product = −15, Sum = −2
Step 2: Numbers = −5 and 3
Step 3: Factor: (x − 5)(x + 3)
Answer: (x − 5)(x + 3)
8. x² + 2x − 3
Step 1: Product = −3, Sum = 2
Step 2: Numbers = 3 and −1
Step 3: Factor: (x + 3)(x − 1)
Answer: (x + 3)(x − 1)
9. x² + x − 20
Step 1: Product = −20, Sum = 1
Step 2: Numbers = 5 and −4
Step 3: Factor: (x + 5)(x − 4)
Answer: (x + 5)(x − 4)
10. x² − x − 20
Step 1: Product = −20, Sum = −1
Step 2: Numbers = −5 and 4
Step 3: Factor: (x − 5)(x + 4)
Answer: (x − 5)(x + 4)
11. 2x² − 22x + 20
Step 1: a × c = 2 × 20 = 40
Step 2: Find two numbers that multiply to 40 and add to −22 → −20 and −2
Step 3: Rewrite: 2x² − 20x − 2x + 20
Step 4: Group: 2x(x − 10) −2(x − 10)
Step 5: Factor: (2x − 2)(x − 10)
Step 6: Factor out common factor: 2(x − 1)(x − 10)
Answer: 2(x − 1)(x − 10)
12. 6a² + 14a + 8
Step 1: a × c = 6 × 8 = 48
Step 2: Find two numbers that multiply to 48 and add to 14 → 6 and 8
Step 3: Rewrite: 6a² + 6a + 8a + 8
Step 4: Group: 6a(a + 1) + 8(a + 1)
Step 5: Factor: (6a + 8)(a + 1)
Step 6: Take out common factor: 2(3a + 4)(a + 1)
Answer: 2(3a + 4)(a + 1)
13. 6v² − 27v + 27
Step 1: Find GCF = 3
= 3(2v² − 9v + 9)
Step 2: Factor inside: 2v² − 6v − 3v + 9
= 3[(2v − 3)(v − 3)]
Answer: 3(2v − 3)(v − 3)
14. 6g² − 15g − 9
Step 1: GCF = 3 → 3(2g² − 5g − 3)
Step 2: Split middle: 2g² − 6g + g − 3
= 3[(2g + 1)(g − 3)]
Answer: 3(2g + 1)(g − 3)
15. 3x² + 19x + 6
a×c = 3×6 = 18, pair: 18 and 1
Split middle: 3x² + 18x + x + 6
= 3x(x + 6) +1(x + 6)
Answer: (3x + 1)(x + 6)
16. 3x² + 17x − 6
a×c = 3×(−6) = −18, pair: 18 and −1
= 3x² + 18x − x − 6
= 3x(x + 6) −1(x + 6)
Answer: (3x − 1)(x + 6)
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