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1. Factor trinomials of the form ax² + bx + c
(i) 2a2 + 11a + 12
(ii) 3a2 + 8a + 4
(iii) 3a2 – 13a + 14
(iv) 4a2 – 7a + 3
(v) 5a2 – 11a – 12
(vi) 7a2 – 15a – 18
Solution:
(i) The given expression 2a2 + 11a + 12
Rewrite the equation
2a2 + 8a + 3a + 12
Group the first two terms and last two terms.
The first two terms are 2a2 + 8a and the second two terms are 3a + 12.
Take 2a common from the first two terms.
2a (a + 4)
Take 3 common from the second two terms.
3 (a + 4)
2a (a + 4) + 3 (a + 4)
Then, take (a + 4) common from the above expression.
(a + 4) (2a + 3)
The final answer is (a + 4) (2a + 3).
(ii) The given expression 3a2 + 8a + 4
Rewrite the equation
3a2 + 6a + 2a + 4
Group the first two terms and last two terms.
The first two terms are 3a2 + 6a and the second two terms are 2a + 4.
Take 3a common from the first two terms.
3a (a + 2)
Take 2 common from the second two terms.
2 (a + 2)
3a (a + 2) + 2 (a + 2)
Then, take (a + 2) common from the above expression.
(a + 2) (3a + 2)
The final answer is (a + 2) (3a + 2).
(iii) The given expression 3a2 – 13a + 14
Rewrite the equation
3a2 – 6a – 7a + 14
Group the first two terms and last two terms.
The first two terms are 3a2 – 6a and the second two terms are – 7a + 14.
Take 3a common from the first two terms.
3a (a – 2)
Take -7 common from the second two terms.
-7 (a – 2)
3a (a – 2) – 7 (a – 2)
Then, take (a – 2) common from the above expression.
(a – 2) (3a – 7)
The final answer is (a – 2) (3a – 7).
(iv) The given expression 4a2 – 7a + 3
Rewrite the equation
4a2 – 4a – 3a + 3
Group the first two terms and last two terms.
The first two terms are 4a2 – 4a and the second two terms are – 3a + 3.
Take 4a common from the first two terms.
4a (a – 1)
Take -3 common from the second two terms.
-3 (a – 1)
4a (a – 1) – 3 (a – 1)
Then, take (a – 1) common from the above expression.
(a – 1) (4a – 3)
The final answer is (a – 1) (4a – 3).
(v) The given expression 5a2 – 11a – 12
Rewrite the equation
5a2 – 15a + 4a – 12
Group the first two terms and last two terms.
The first two terms are 5a2 – 15a and the second two terms are 4a – 12.
Take 4a common from the first two terms.
5a (a – 3)
Take 4 common from the second two terms.
4 (a – 3)
5a (a – 3) + 4 (a – 3)
Then, take (a – 3) common from the above expression.
(a – 3) (5a + 4)
The final answer is (a – 3) (5a + 4).
(vi) The given expression 7a2 – 15a – 18
Rewrite the equation
7a2 – 21a + 6a – 18
Group the first two terms and last two terms.
The first two terms are 7a2 – 21a and the second two terms are 6a – 18.
Take 7a common from the first two terms.
7a (a – 3)
Take 6 common from the second two terms.
6 (a – 3)
7a (a – 3) + 6 (a – 3)
Then, take (a – 3) common from the above expression.
(a – 3) (7a + 6)
The final answer is (a – 3) (7a + 6).
2. Factor trinomials
(i) 7a2 + 6a – 1
(ii) 9a2 + 35a – 4
(iii) 2a2 – 5a + 3
(iv) 7a – 6 – 2a2
(v) 11a2 – 54a + 63
(vi) a2 + 2a – 3
Solution:
(i) The given expression 7a2 + 6a – 1
Rewrite the equation
7a2 + 7a – a – 1
Group the first two terms and last two terms.
The first two terms are 7a2 + 7a and the second two terms are – a – 1.
Take 7a common from the first two terms.
7a (a + 1)
Take -1 common from the second two terms.
-1 (a + 1)
7a (a + 1) – 1 (a + 1)
Then, take (a + 1) common from the above expression.
(a + 1) (7a – 1)
The final answer is (a + 1) (7a – 1).
(ii) The given expression 9a2 + 35a – 4
Rewrite the equation
9a2 + 36a – a – 4
Group the first two terms and last two terms.
The first two terms are 9a2 + 36a and the second two terms are – a – 4.
Take 9a common from the first two terms.
9a (a + 4)
Take -1 common from the second two terms.
-1 (a + 4)
9a (a + 4) – 1 (a + 4)
Then, take (a + 4) common from the above expression.
(a + 4) (9a – 1)
The final answer is (a + 4) (9a – 1).
(iii) The given expression 2a2 – 5a + 3
Rewrite the equation
2a2 – 2a – 3a + 3
Group the first two terms and last two terms.
The first two terms are 2a2 – 2a and the second two terms are – 3a + 3.
Take 2a common from the first two terms.
2a (a – 1)
Take -3 common from the second two terms.
-3 (a – 1)
2a (a – 1) – 3 (a – 1)
Then, take (a – 1) common from the above expression.
(a – 1) (2a – 3)
The final answer is (a – 1) (2a – 3).
(iv) The given expression 7a – 6 – 2a2
Rewrite the equation
– 2a2 + 4a + 3a – 6
Group the first two terms and last two terms.
The first two terms are – 2a2 + 4a and the second two terms are 3a – 6.
Take -2a common from the first two terms.
-2a (a – 2)
Take 3 common from the second two terms.
3 (a – 2)
-2a (a – 2) + 3 (a – 2)
Then, take (a – 2) common from the above expression.
(a – 2) (-2a + 3)
The final answer is (a – 2) (-2a + 3).
(v) The given expression 11a2 – 54a + 63
Rewrite the equation
11a2 – 33a – 21a + 63
Group the first two terms and last two terms.
The first two terms are 11a2 – 33a and the second two terms are – 21a + 63.
Take 11a common from the first two terms.
11a (a – 3)
Take -21 common from the second two terms.
-21 (a – 3)
11a (a – 3) – 21 (a – 3)
Then, take (a – 3) common from the above expression.
(a – 3) (11a – 21)
The final answer is (a – 3) (11a – 21).
(vi) The given expression a2 + 2a – 3
Rewrite the equation
a2 + 3a – a – 3
Group the first two terms and last two terms.
The first two terms are a2 + 3a and the second two terms are – a – 3.
Take a common from the first two terms.
a (a + 3)
Take -1 common from the second two terms.
-1 (a + 3)
a (a + 3) – 1 (a + 3)
Then, take (a + 3) common from the above expression.
(a + 3) (a – 1)
The final answer is (a + 3) (a – 1).
3. Factorize the quadratic expression
(i) 2x2 + 5x + 2
(ii) 3a2 + 14a + 8
(iii) 2x2 + 7x + 6
(iv) 6a2 – a – 15
(v) 9s2 – s – 8
(vi) 12 + a – 6a2
(vii) 6 + 5x – 6x2
(viii) a2 + 8a – 105
Solution:
(i) The given expression 2x2 + 5x + 2
Rewrite the equation
2x2 + 4x + x + 2
Group the first two terms and last two terms.
The first two terms are 2x2 + 4x and the second two terms are x + 2.
Take 2x common from the first two terms.
2x (x + 2)
Take 1 common from the second two terms.
1 (x + 2)
2x (x + 2) + 1 (x + 2)
Then, take (x + 2) common from the above expression.
(x + 2) (2x + 1)
The final answer is (x + 2) (2x + 1).
(ii) The given expression 3a2 + 14a + 8
Rewrite the equation
3a2 + 12a + 2a + 8
Group the first two terms and last two terms.
The first two terms are 3a2 + 12a and the second two terms are 2a + 8.
Take 3a common from the first two terms.
3a (a + 4)
Take 2 common from the second two terms.
2 (a + 4)
3a (a + 4) + 2 (a + 4)
Then, take (a + 4) common from the above expression.
(a + 4) (3a + 2)
The final answer is (a + 4) (3a + 2).
(iii) The given expression 2x2 + 7x + 6
Rewrite the equation
2x2 + 4x + 3x + 6
Group the first two terms and last two terms.
The first two terms are 2x2 + 4x and the second two terms are 3x + 6.
Take 2x common from the first two terms.
2x (x + 2)
Take 3 common from the second two terms.
3 (x + 2)
2x (x + 2) + 3 (x + 2)
Then, take (x + 2) common from the above expression.
(x + 2) (2x + 3)
The final answer is (x + 2) (2x + 3).
(iv) The given expression 6a2 – a – 15
Rewrite the equation
6a2 + 9a -10a – 15
Group the first two terms and last two terms.
The first two terms are 6a2 + 9a and the second two terms are -10a – 15.
Take 3a common from the first two terms.
3a (2a + 3)
Take -5 common from the second two terms.
-5 (2a + 3)
3a (2a + 3) -5 (2a + 3)
Then, take (2a + 3) common from the above expression.
(2a + 3) (3a – 5)
The final answer is (2a + 3) (3a – 5).
(v) The given expression 9s2 – s – 8
Rewrite the equation
9s2 – 9s + 8s – 8
Group the first two terms and last two terms.
The first two terms are 9s2 – 9s and the second two terms are 8s – 8.
Take 9s common from the first two terms.
9s (s – 1)
Take 8 common from the second two terms.
8 (s – 1)
9s (s – 1) + 8 (s – 1)
Then, take (s – 1) common from the above expression.
(s – 1) (9s + 8)
The final answer is (s – 1) (9s + 8).
(vi) The given expression 12 + a – 6a2
Rewrite the equation
– 6a2 + 9a -8a + 12
Group the first two terms and last two terms.
The first two terms are – 6a2 + 9a and the second two terms are -8a + 12 .
Take -3a common from the first two terms.
-3a (2a – 3)
Take -4 common from the second two terms.
-4 (2a – 3)
-3a (2a – 3) – 4 (2a – 3)
Then, take (2a – 3) common from the above expression.
(2a – 3) (-3a – 4)
The final answer is (2a – 3) (-3a – 4).
(vii) The given expression 6 + 5x – 6x2
Rewrite the equation
– 6x2 + 9x – 4x +6
Group the first two terms and last two terms.
The first two terms are – 6x2 + 9x and the second two terms are – 4x +6.
Take -3x common from the first two terms.
-3x (2x – 3)
Take -2 common from the second two terms.
-2 (2x – 3)
-3x (2x – 3) -2 (2x – 3)
Then, take (2x – 3) common from the above expression.
(2x – 3) (-3x – 2)
The final answer is (2x – 3) (-3x – 2).
(viii) The given expression a2 + 8a – 105
Rewrite the equation
a2 + 15a – 7a – 105
Group the first two terms and last two terms.
The first two terms are a2 + 15a and the second two terms are – 7a – 105.
Take a common from the first two terms.
a (a + 15)
Take -7 common from the second two terms.
-7 (a + 15)
a (a + 15) – 7 (a + 15)
Then, take (a + 15) common from the above expression.
(a + 15) (a – 7)
The final answer is (a + 15) (a – 7).