Math Mantra Nepal – Factorization Class 8 (Page 156)
⚠️ Note: Questions from A to S (except C, F, I, L, O, R, T) should be practiced by yourselves. Questions 3 and 4 follow the same rules. Comment on YouTube if you face any issues!
Class 8 – Mathematics (Factorization)
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1. Rules & Concepts
2. CDC Questions (Q3)
3. Difficult Questions (Q4 & Q5)
(तल दिइएको युट्युब भिडियो हेरेर मिड-टर्म स्प्लिटिङ विधि अझ सजिलैसँग सिक्नुहोस्।)
Standard Form of Quadratic Expression: Splitting Midterms
Standard Quadratic Expression Form:
$$ax^2 + bx + c$$
(यहाँ मध्य पद $bx$ लाई दुईवटा यस्ता पदहरूमा फुटाइन्छ जसको गुणनफल $a \times c$ र जोडफल/घटाउफल $b$ सँग बराबर हुन्छ।)
(मध्य पदलाई फुटाउनु अघि, यी महत्त्वपूर्ण नियम तथा ट्रिकहरू ध्यान दिएर सिकौँ।)
1) Rules Based on Product and Sum
(यदि अन्तिम पदको चिह्न (+) छ भने हामीले दुईवटा यस्ता संख्याहरू खोज्नुपर्छ जसको गुणनफल Product सँग र जोडफल Sum सँग बराबर होस्।)
Example Trick: Think about two numbers whose Product is 10 and Sum is 7 .
Factor Pair
Product Check
Sum Check
Decision
1 and 10
1 × 10 = 10
10 + 1 = 11
✘ Rejected (Cross)
2 and 5
2 × 5 = 10
5 + 2 = 7
✔ Accepted (Tick)
👉 So, the two required numbers are 5 and 2 . (Write larger number first!)
2) Rules Based on Product and Difference
(यदि अन्तिम पदको चिह्न (-) छ भने हामीले दुईवटा यस्ता संख्याहरू खोज्नुपर्छ जसको गुणनफल Product सँग र घटाउफल Difference सँग बराबर होस्।)
Example Trick: Think about two numbers whose Product is 10 and Difference is 3 .
Factor Pair
Product Check
Difference Check
Decision
1 and 10
1 × 10 = 10
10 – 1 = 9
✘ Rejected (Cross)
2 and 5
2 × 5 = 10
5 – 2 = 3
✔ Accepted (Tick)
👉 So, the two required numbers are 5 and 2 . (Reason: $+5x – 2x = +3x$)
📖 CDC Page 156 Question 3 Solutions
(पाठ्यक्रम विकास केन्द्र CDC कक्षा ८ प्रश्न नं. ३ का विस्तृत हलहरू)
Page 156
Question 3 (C)
Solution:
The given expression is:
$$x^2 – 5x + 6$$
Step 1: Split the middle term
$= x^2 – 3x – 2x + 6$
Step 2: Take common factors from groups
$= x(x – 3) – 2(x – 3)$
Step 3: Combine common binomials
$= (x – 3)(x – 2)$
Rough Work Space
Condition:
Product = 6
Sum = 5
Pairs: 3 × 2 = 6
Reason: $-3x – 2x = -5x$ ✔
Numbers: 3, 2
Note: Plus/minus sign at the third term can be adjusted.
Question 3 (F)
Solution:
The given expression is:
$$a^2 – 3a + 2$$
Step 1: Split the middle term
$= a^2 – 2a – 1a + 2$
Step 2: Take common factors
$= a(a – 2) – 1(a – 2)$
Step 3: Combine common binomials
$= (a – 2)(a – 1)$
Rough Work Space
Condition:
Product = 2
Sum = 3
Pairs: 2 × 1 = 2
Reason: $-2a – 1a = -3a$ ✔
Numbers: 2, 1
Note: Plus/minus sign at the third term can be adjusted.
Question 3 (I)
Solution:
The given expression is:
$$b^2 + 13b + 42$$
Step 1: Split the middle term
$= b^2 + 7b + 6b + 42$
Step 2: Take common factors
$= b(b + 7) + 6(b + 7)$
Step 3: Combine common binomials
$= (b + 7)(b + 6)$
Rough Work Space
Condition:
Product = 42
Sum = 13
Pairs: 7 × 6 = 42
Reason: $+7b + 6b = +13b$ ✔
Numbers: 7, 6
Note: Plus/minus sign at the third term can be adjusted.
Question 3 (L)
Solution:
The given expression is:
$$x^2 – 15x + 56$$
Step 1: Split the middle term
$= x^2 – 8x – 7x + 56$
Step 2: Take common factors
$= x(x – 8) – 7(x – 8)$
Step 3: Combine common binomials
$= (x – 8)(x – 7)$
Rough Work Space
Condition:
Product = 56
Sum = 15
Pairs: 8 × 7 = 56
Reason: $-8x – 7x = -15x$ ✔
Numbers: 8, 7
Note: Plus/minus sign at the third term can be adjusted.
Question 3 (O)
Solution:
The given expression is:
$$b^2 – 12b + 36$$
Step 1: Split the middle term
$= b^2 – 6b – 6b + 36$
Step 2: Take common factors
$= b(b – 6) – 6(b – 6)$
Step 3: Combine common binomials
$= (b – 6)(b – 6) \quad \text{or } (b – 6)^2$$
Rough Work Space
Condition:
Product = 36
Sum = 12
Pairs: 6 × 6 = 36
Reason: $-6b – 6b = -12b$ ✔
Numbers: 6, 6
Note: Plus/minus sign at the third term can be adjusted.
Question 3 (R)
Solution:
The given expression is:
$$x^2 – 23x + 102$$
Step 1: Split the middle term
$= x^2 – 17x – 6x + 102$
Step 2: Take common factors
$= x(x – 17) – 6(x – 17)$
Step 3: Combine common binomials
$= (x – 17)(x – 6)$
Rough Work Space
Condition:
Product = 102
Sum = 23
Pairs: 17 × 6 = 102
Reason: $-17x – 6x = -23x$ ✔
Numbers: 17, 6
Note: Plus/minus sign at the third term can be adjusted.
Question 3 (T)
Solution:
The given expression is:
$$(x+y)^2 – 15(x+y) + 36$$
Step 1: Substitute $(x + y) = a$
$= a^2 – 15a + 36$
Step 2: Split middle term
$= a^2 – 12a – 3a + 36$
Step 3: Factorize by grouping
$= a(a – 12) – 3(a – 12) = (a – 12)(a – 3)$
Step 4: Substitute back $a = (x + y)$
$= (x + y – 12)(x + y – 3)$
Substitution Rough Work
Let $(x + y) = a$
Product = 36
Sum = 15
Pairs: 12 × 3 = 36
Reason: $-12a – 3a = -15a$ ✔
Numbers: 12, 3
Note: Plus/minus sign at the third term can be adjusted.
🔥 Difficult Questions Section (Page 156: Question 4 & Question 5)
(विशेष र कडा खालका गणितीय अभिव्यक्तिका प्रश्नहरू र तिनको विस्तृत हल)
📌 Question 4: Re-solve into Factors
Question 4 (a)
Solution:
The given expression is:
$$x^2 + 4x – 21$$
Step 1: Split the middle term
$= x^2 + 7x – 3x – 21$
Step 2: Take common factors
$= x(x + 7) – 3(x + 7)$
Step 3: Combine common binomials
$= (x + 7)(x – 3)$
Rough Work Space
Condition:
Product = 21
Difference = 4
Pairs: 7 × 3 = 21
Reason: $+7x – 3x = +4x$ ✔
Numbers: 7, 3
Question 4 (d)
Solution:
The given expression is:
$$y^2 – 6y – 27$$
Step 1: Split the middle term
$= y^2 – 9y + 3y – 27$
Step 2: Take common factors
$= y(y – 9) + 3(y – 9)$
Step 3: Combine common binomials
$= (y – 9)(y + 3)$
Rough Work Space
Condition:
Product = 27
Difference = 6
Pairs: 9 × 3 = 27
Reason: $-9y + 3y = -6y$ ✔
Numbers: 9, 3
Question 4 (g)
Solution:
The given expression is:
$$a^2 – a – 132$$
Step 1: Split the middle term
$= a^2 – 12a + 11a – 132$
Step 2: Take common factors
$= a(a – 12) + 11(a – 12)$
Step 3: Combine common binomials
$= (a – 12)(a + 11)$
Rough Work Space
Condition:
Product = 132
Difference = 1
Pairs: 12 × 11 = 132
Reason: $-12a + 11a = -1a$ ✔
Numbers: 12, 11
Question 4 (l)
Solution:
The given expression is:
$$x^2 + xy – 240y^2$$
Step 1: Split the middle term
$= x^2 + 16xy – 15xy – 240y^2$
Step 2: Take common factors
$= x(x + 16y) – 15y(x + 16y)$
Step 3: Combine common binomials
$= (x + 16y)(x – 15y)$
Rough Work Space
Condition:
Product = 240
Difference = 1
Pairs: 16 × 15 = 240
Reason: $+16xy – 15xy = +1xy$ ✔
Numbers: 16, 15
Question 4 (m)
Solution:
The given expression is:
$$35 – 2x – x^2$$
Step 1: Split the middle term
$= 35 – 7x + 5x – x^2$
Step 2: Take common factors
$= 7(5 – x) + x(5 – x)$
Step 3: Combine common binomials
$= (5 – x)(7 + x)$
Rough Work Space
Condition:
Product = 35
Difference = 2
Pairs: 7 × 5 = 35
Reason: $-7x + 5x = -2x$ ✔
Numbers: 7, 5
Note: Questions (n) and (o) follow the exact same method as (m).
Question 4 (p)
Solution:
The given expression is:
$$(a+b)^2 + 5(a+b) – 36$$
Step 1: Substitute $(a + b) = x$
$= x^2 + 5x – 36$
Step 2: Split middle term
$= x^2 + 9x – 4x – 36$
Step 3: Factorize by grouping
$= x(x + 9) – 4(x + 9) = (x + 9)(x – 4)$
Step 4: Substitute back $x = (a + b)$
$= (a + b + 9)(a + b – 4)$
Substitution Rough Work
Let $(a + b) = x$
Product = 36
Difference = 5
Pairs: 9 × 4 = 36
Reason: $+9x – 4x = +5x$ ✔
Numbers: 9, 4
📌 Question 5: Factorize Advanced Quadratic Expressions ($ax^2 + bx + c$)
Question 5 (a)
Solution:
The given expression is:
$$3x^2 + 5x + 2$$
Step 1: Split the middle term
$= 3x^2 + 3x + 2x + 2$
Step 2: Take common factors
$= 3x(x + 1) + 2(x + 1)$
Step 3: Combine common binomials
$= (x + 1)(3x + 2)$
Rough Work Space
Product = $3 \times 2 = 6$
Sum = 5
Pairs: 3 × 2 = 6
Reason: $+3x + 2x = +5x$ ✔
Numbers: 3, 2
Question 5 (d)
Solution:
The given expression is:
$$4a^2 – 8a + 3$$
Step 1: Split the middle term
$= 4a^2 – 6a – 2a + 3$
Step 2: Take common factors
$= 2a(2a – 3) – 1(2a – 3)$
Step 3: Combine common binomials
$= (2a – 3)(2a – 1)$
Rough Work Space
Product = $4 \times 3 = 12$
Sum = 8
Pairs: 6 × 2 = 12
Reason: $-6a – 2a = -8a$ ✔
Numbers: 6, 2
Question 5 (g)
Solution:
The given expression is:
$$5x^2 – 14x – 3$$
Step 1: Split the middle term
$= 5x^2 – 15x + 1x – 3$
Step 2: Take common factors
$= 5x(x – 3) + 1(x – 3)$
Step 3: Combine common binomials
$= (x – 3)(5x + 1)$
Rough Work Space
Product = $5 \times 3 = 15$
Difference = 14
Pairs: 15 × 1 = 15
Reason: $-15x + 1x = -14x$ ✔
Numbers: 15, 1
Question 5 (j)
Solution:
The given expression is:
$$6b^2 – 4b – 10$$
Step 1: Take common factor $2$ first
$= 2(3b^2 – 2b – 5)$
Step 2: Split the middle term inside bracket
$= 2(3b^2 – 5b + 3b – 5)$
Step 3: Factorize inside bracket
$= 2[b(3b – 5) + 1(3b – 5)]$
$= 2(3b – 5)(b + 1)$
Rough Work Space
Take Common First: $2$
For $3b^2 – 2b – 5$:
Product = $3 \times 5 = 15$
Difference = 2
Pairs: 5 × 3 = 15
Reason: $-5b + 3b = -2b$ ✔
Numbers: 5, 3
Question 5 (m)
Solution:
The given expression is:
$$16a^2 + 24ab + 9b^2$$
Method 1: Perfect Square Method $(a+b)^2$
$= (4a)^2 + 2 \cdot (4a) \cdot (3b) + (3b)^2$
$= (4a + 3b)^2$
$= (4a + 3b)(4a + 3b)$
Method 2: Mid-Term Splitting Method
$= 16a^2 + 12ab + 12ab + 9b^2$
$= 4a(4a + 3b) + 3b(4a + 3b)$
$= (4a + 3b)(4a + 3b)$
Rough Work Space
For Mid-Term Splitting:
Product = $16 \times 9 = 144$
Sum = 24
Pairs: 12 × 12 = 144
Reason: $+12ab + 12ab = +24ab$ ✔
Numbers: 12, 12
Question 5 (p)
Solution:
The given expression is:
$$6p^2q + 30pq + 36q$$
Step 1: Take common factor $6q$ first
$= 6q(p^2 + 5p + 6)$
Step 2: Split middle term inside bracket
$= 6q(p^2 + 3p + 2p + 6)$
Step 3: Factorize inside bracket
$= 6q[p(p + 3) + 2(p + 3)]$
$= 6q(p + 3)(p + 2)$
Rough Work Space
Take Common First: $6q$
For $p^2 + 5p + 6$:
Product = 6
Sum = 5
Pairs: 3 × 2 = 6
Reason: $+3p + 2p = +5p$ ✔
Numbers: 3, 2
Question 5 (s)
Solution:
The given expression is:
$$4 + 17x – 15x^2$$
Step 1: Split the middle term
$= 4 + 20x – 3x – 15x^2$
Step 2: Take common factors
$= 4(1 + 5x) – 3x(1 + 5x)$
Step 3: Combine common binomials
$= (1 + 5x)(4 – 3x)$
Rough Work Space
Product = $4 \times 15 = 60$
Difference = 17
Pairs: 20 × 3 = 60
Reason: $+20x – 3x = +17x$ ✔
Numbers: 20, 3
Question 5 (u)
Solution:
The given expression is:
$$28 – 31b – 5b^2$$
Step 1: Split the middle term
$= 28 – 35b + 4b – 5b^2$
Step 2: Take common factors
$= 7(4 – 5b) + b(4 – 5b)$
Step 3: Combine common binomials
$= (4 – 5b)(7 + b)$
Rough Work Space
Product = $28 \times 5 = 140$
Difference = 31
Pairs: 35 × 4 = 140
Reason: $-35b + 4b = -31b$ ✔
Numbers: 35, 4
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