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1. Factorize the quadratic trinomials
(i) a2 + 5a + 6
(ii) a2 + 10a + 24
(iii) a2 + 12a + 27
(iv) a2 + 15a + 56
(v) a2 + 19a + 60
(vi) a2 + 13a + 40
(vii) a2 – 10a + 24
(viii) a2 – 23a + 42
(ix) a2 – 17a + 16
(x) a2 – 21a + 90
Solution:
(i) The Given expression is a2 + 5a + 6.
By comparing the given expression a2 + 5a + 6 with the basic expression x^2 + ax + b.
Here, a = 1, b = 5, and c = 6.
The sum of two numbers is m + n = b = 5 = 3 + 2.
The product of two number is m * n = a * c = 1 * (6) = 6 = 3 * 2
From the above two instructions, we can write the values of two numbers m and n as 3 and 2.
Then, a2 + 5a + 6 = a2 + 3a + 2a + 6.
= a (a+ 3) + 2(a + 3).
Factor out the common terms.
(a + 3) (a + 2)
Then, a2 + 5a + 6 = (a + 3) (a + 2).
(ii) The Given expression is a2 + 10a + 24.
By comparing the given expression a2 + 5a + 6 with the basic expression x^2 + ax + b.
Here, a = 1, b = 10, and c = 24.
The sum of two numbers is m + n = b = 10 = 6 + 4.
The product of two number is m * n = a * c = 1 * (24) = 24 = 6 * 4
From the above two instructions, we can write the values of two numbers m and n as 6 and 4.
Then, a2 + 10a + 24 = a2 + 6a + 4a + 24.
= a (a+ 6) + 4(a + 6).
Factor out the common terms.
(a + 6) (a + 4)
Then, a2 + 10a + 24 = (a + 6) (a + 4).
(iii) The Given expression is a2 + 12a + 27.
By comparing the given expression a2 + 12a + 27 with the basic expression x^2 + ax + b.
Here, a = 1, b = 12, and c = 27.
The sum of two numbers is m + n = b = 12 = 9 + 3.
The product of two number is m * n = a * c = 1 * (27) = 27 = 9 * 3
From the above two instructions, we can write the values of two numbers m and n as 9 and 3.
Then, a2 + 12a + 27 = a2 + 9a + 3a + 27.
= a (a+ 9) + 3(a + 9).
Factor out the common terms.
(a + 9) (a + 3)
Then, a2 + 12a + 27 = (a + 9) (a + 3).
(iv) The Given expression is a2 + 15a + 56.
By comparing the given expression a2 + 15a + 56 with the basic expression x^2 + ax + b.
Here, a = 1, b = 15, and c = 56.
The sum of two numbers is m + n = b = 15 = 8 + 7.
The product of two number is m * n = a * c = 1 * (56) = 56 = 8 * 7
From the above two instructions, we can write the values of two numbers m and n as 8 and 7.
Then, a2 + 15a + 56 = a2 + 8a + 7a + 56.
= a (a+ 8) + 7(a + 8).
Factor out the common terms.
(a + 8) (a + 7)
Then, a2 + 15a + 56 = (a + 8) (a + 7).
(v) The Given expression is a2 + 19a + 60.
By comparing the given expression a2 + 19a + 60 with the basic expression x^2 + ax + b.
Here, a = 1, b = 19, and c = 60.
The sum of two numbers is m + n = b = 19 = 15 + 4.
The product of two number is m * n = a * c = 1 * (60) = 60 = 15 * 4
From the above two instructions, we can write the values of two numbers m and n as 15 and 4.
Then, a2 + 19a + 60 = a2 + 15a + 4a + 60.
= a (a+ 15) + 4(a + 15).
Factor out the common terms.
(a + 15) (a + 4)
Then, a2 + 19a + 60 = (a + 15) (a + 4).
(vi) The Given expression is a2 + 13a + 40.
By comparing the given expression a2 + 13a + 40 with the basic expression x^2 + ax + b.
Here, a = 1, b = 13, and c = 40.
The sum of two numbers is m + n = b = 13 = 8 + 5.
The product of two number is m * n = a * c = 1 * (40) = 40 = 8 * 5
From the above two instructions, we can write the values of two numbers m and n as 8 and 5.
Then, a2 + 13a + 40 = a2 + 8a + 5a + 40.
= a (a + 8) + 5(a + 8).
Factor out the common terms.
(a + 8) (a + 5)
Then, a2 + 13a + 40 = (a + 8) (a + 5).
(vii) The Given expression is a2 – 10a + 24.
By comparing the given expression a2 – 10a + 24 with the basic expression x^2 + ax + b.
Here, a = 1, b = -10, and c = 24.
The sum of two numbers is m + n = b = -10 = -6 – 4.
The product of two number is m * n = a * c = 1 * (24) = 24 = -6 * -4
From the above two instructions, we can write the values of two numbers m and n as -6 and -4.
Then, a2 – 10a + 24 = a2 – 6a – 4a + 24.
= a (a – 6) – 4(a – 6).
Factor out the common terms.
(a – 6) (a – 4)
Then, a2 – 10a + 24 = (a – 6) (a – 4).
(viii) The Given expression is a2 – 23a + 42.
By comparing the given expression a2 – 23a + 42 with the basic expression x^2 + ax + b.
Here, a = 1, b = -23, and c = 42.
The sum of two numbers is m + n = b = -23 = -21 – 2.
The product of two number is m * n = a * c = 1 * (42) = 42 = -21 * -2
From the above two instructions, we can write the values of two numbers m and n as -21 and -2.
Then, a2 – 23a + 42 = a2 – 21a – 2a + 42.
= a (a – 21) – 2(a – 21).
Factor out the common terms.
(a – 21) (a – 2)
Then, a2 – 23a + 42 = (a – 21) (a – 2).
(ix) The Given expression is a2 – 17a + 16.
By comparing the given expression a2 – 17a + 16 with the basic expression x^2 + ax + b.
Here, a = 1, b = -17, and c = 16.
The sum of two numbers is m + n = b = -17 = -16 – 1.
The product of two number is m * n = a * c = 1 * (16) = 16 = -16 * -1
From the above two instructions, we can write the values of two numbers m and n as -16 and -1.
Then, a2 – 17a + 16 = a2 – 16a -a + 16.
= a (a – 16) – 1(a – 16).
Factor out the common terms.
(a – 16) (a – 1)
Then, a2 – 17a + 16 = (a – 16) (a – 1).
(x) The Given expression is a2 – 21a + 90.
By comparing the given expression a2 – 21a + 90 with the basic expression x^2 + ax + b.
Here, a = 1, b = -21, and c = 90.
The sum of two numbers is m + n = b = -21 = -15 – 6.
The product of two number is m * n = a * c = 1 * (90) = 90 = -15 * -6
From the above two instructions, we can write the values of two numbers m and n as -15 and -6.
Then, a2 – 21a + 90 = a2 – 15a – 6a + 90.
= a (a – 15) – 6(a – 15).
Factor out the common terms.
(a – 15) (a – 6)
Then, a2 – 21a + 90 = (a – 15) (a – 6).
2. Factorize the expressions completely
(i) a2 – 22a + 117
(ii) a2 – 9a + 20
(iii) a2 + a – 132
(iv) a2 + 5a – 104
(v) b2 + 7b – 144
(vi) c2 + 19c – 150
(vii) b2 + b – 72
(viii) a2 + 6a – 91
(ix) a2 – 4a -77
(x) a2 – 6a – 135
Solution:
(i) The Given expression is a2 – 22a + 117.
By comparing the given expression a2 – 22a + 117 with the basic expression x^2 + ax + b.
Here, a = 1, b = -22, and c = 117.
The sum of two numbers is m + n = b = -22 = -13 – 9.
The product of two number is m * n = a * c = 1 * (117) = 117 = -13 * -9
From the above two instructions, we can write the values of two numbers m and n as -13 and -9.
Then, a2 – 22a + 117 = a2 – 13a – 9a + 117.
= a (a – 13) – 9(a – 13).
Factor out the common terms.
(a – 13) (a – 9)
Then, a2 – 22a + 117 = (a – 13) (a – 9).
(ii) The Given expression is a2 – 9a + 20.
By comparing the given expression a2 – 9a + 20 with the basic expression x^2 + ax + b.
Here, a = 1, b = -9, and c = 20.
The sum of two numbers is m + n = b = -9 = -5 – 4.
The product of two number is m * n = a * c = 1 * (20) = 20 = -5 * -4
From the above two instructions, we can write the values of two numbers m and n as -5 and -4.
Then, a2 – 9a + 20 = a2 – 5a -4a + 20.
= a (a – 5) – 4(a – 5).
Factor out the common terms.
(a – 5) (a – 4)
Then, a2 – 9a + 20 = (a – 5) (a – 4).
(iii) The Given expression is a2 + a – 132.
By comparing the given expression a2 + a – 132 with the basic expression x^2 + ax + b.
Here, a = 1, b = 1, and c = -132.
The sum of two numbers is m + n = b = 1 = 12 – 11.
The product of two number is m * n = a * c = 1 * (-132) = -132 = 12 * -11
From the above two instructions, we can write the values of two numbers m and n as 12 and -11.
Then, a2 + a – 132 = a2 + 12a – 11a – 132.
= a (a + 12) – 11(a + 12).
Factor out the common terms.
(a + 12) (a – 11)
Then, a2 + a – 132 = (a + 12) (a – 11).
(iv) The Given expression is a2 + 5a – 104.
By comparing the given expression a2 + 5a – 104 with the basic expression x^2 + ax + b.
Here, a = 1, b = 5, and c = -104.
The sum of two numbers is m + n = b = 5 = 13 – 8.
The product of two number is m * n = a * c = 1 * (-104) = -104 = 13 * -8
From the above two instructions, we can write the values of two numbers m and n as 13 and -8.
Then, a2 + 5a – 104 = a2 + 13a – 8a – 104.
= a (a + 13) – 8(a + 13).
Factor out the common terms.
(a + 13) (a – 8)
Then, a2 + 5a – 104 = (a + 13) (a – 8).
(v) The Given expression is b2 + 7b – 144.
By comparing the given expression b2 + 7b – 144 with the basic expression x^2 + ax + b.
Here, a = 1, b = 7, and c = -144.
The sum of two numbers is m + n = b = 7 = 16 – 9.
The product of two number is m * n = a * c = 1 * (-144) = -144 = 16 * -9
From the above two instructions, we can write the values of two numbers m and n as 16 and -9.
Then, b2 + 7b – 144 = b2 + 16b – 9b – 144.
= b (b + 16) – 9(b + 16).
Factor out the common terms.
(b + 16) (b – 9)
Then, b2 + 7b – 144 = (b + 16) (b – 9).
(vi) The Given expression is c2 + 19c – 150.
By comparing the given expression c2 + 19c – 150 with the basic expression x^2 + ax + b.
Here, a = 1, b = 19, and c = -150.
The sum of two numbers is m + n = b = 19 = 25 – 6.
The product of two number is m * n = a * c = 1 * (-150) = -150 = 25 * -6
From the above two instructions, we can write the values of two numbers m and n as 25 and -6.
Then, c2 + 19c – 150 = c2 + 25c – 6c – 150.
= c (c + 25) – 6(c + 25).
Factor out the common terms.
(c + 25) (c – 6)
Then, c2 + 19c – 150 = (c + 25) (c – 6).
(vii) The Given expression is b2 + b – 72.
By comparing the given expression b2 + b – 72 with the basic expression x^2 + ax + b.
Here, a = 1, b = 1, and c = -72.
The sum of two numbers is m + n = b = 1 = 9 – 8.
The product of two number is m * n = a * c = 1 * (-72) = -72 = 9 * -8
From the above two instructions, we can write the values of two numbers m and n as 9 and -8.
Then, b2 + b – 72 = b2 + 9b – 8b – 72.
= b (b + 9) – 8(b + 9).
Factor out the common terms.
(b + 9) (b – 8)
Then, b2 + b – 72 = (b + 9) (b – 8).
(viii) The Given expression is a2 + 6a – 91.
By comparing the given expression a2 + 6a – 91 with the basic expression x^2 + ax + b.
Here, a = 1, b = 6, and c = -91.
The sum of two numbers is m + n = b = 6 = 13 – 7.
The product of two number is m * n = a * c = 1 * (-91) = -91 = 13 * -7
From the above two instructions, we can write the values of two numbers m and n as 13 and -7.
Then, a2 + 6a – 91 = a2 + 13a – 7a – 91.
= a (a + 13) – 7(a + 13).
Factor out the common terms.
(a + 13) (a – 7)
Then, a2 + 6a – 91 = (a + 13) (a – 7).
(ix) The Given expression is a2 – 4a -77.
By comparing the given expression a2 – 4a -77 with the basic expression x^2 + ax + b.
Here, a = 1, b = -4, and c = -77.
The sum of two numbers is m + n = b = -4 = -11 + 7.
The product of two number is m * n = a * c = 1 * (-77) = -77 = -11 * 7
From the above two instructions, we can write the values of two numbers m and n as -11 and 7.
Then, a2 – 4a -77 = a2 – 11a + 7a -77.
= a (a – 11) + 7(a – 11).
Factor out the common terms.
(a – 11) (a + 7)
Then, a2 – 4a -77 = (a – 11) (a + 7).
(x) The Given expression is a2 – 6a – 135.
By comparing the given expression a2 – 6a – 135 with the basic expression x^2 + ax + b.
Here, a = 1, b = -6, and c = -135.
The sum of two numbers is m + n = b = -6 = -15 + 9.
The product of two number is m * n = a * c = 1 * (-135) = -135 = -15 * 9
From the above two instructions, we can write the values of two numbers m and n as -15 and 9.
Then, a2 – 6a – 135 = a2 – 15a + 9a – 135.
= a (a – 15) + 9(a – 15).
Factor out the common terms.
(a – 15) (a + 9)
Then, a2 – 6a – 135 = (a – 15) (a + 9).
3. Factor by splitting the middle term
(i) a2 – 11a – 42
(ii) a2 – 12a – 45
(iii) a2 – 7a – 30
(iv) a2 – 5a – 24
(v) 3a2 + 10a + 8
(vi) 3a2 + 14a + 8
(vii) 2a2 + a – 45
(viii) 6a2 + 11a – 10
(ix) 3a2 – 10a + 8
(x) 2a2 – 17a – 30
Solution:
(i) The Given expression is a2 – 11a – 42.
By comparing the given expression a2 – 11a – 42 with the basic expression x^2 + ax + b.
Here, a = 1, b = -11, and c = -42.
The sum of two numbers is m + n = b = -11 = -14 + 3.
The product of two number is m * n = a * c = 1 * (-42) = -42 = -14 * 3
From the above two instructions, we can write the values of two numbers m and n as -14 and 3.
Then, a2 – 11a – 42 = a2 – 14a + 3a- 42.
= a (a – 14) + 3(a – 14).
Factor out the common terms.
(a – 14) (a + 3)
Then, a2 – 11a – 42 = (a – 14) (a + 3).
(ii) The Given expression is a2 – 12a – 45.
By comparing the given expression a2 – 12a – 45 with the basic expression x^2 + ax + b.
Here, a = 1, b = -12, and c = -45.
The sum of two numbers is m + n = b = -12 = -15 + 3.
The product of two number is m * n = a * c = 1 * (-45) = -45 = -15 * 3
From the above two instructions, we can write the values of two numbers m and n as -15 and 3.
Then, a2 – 12a – 45 = a2 – 15a + 3a- 45.
= a (a – 15) + 3(a – 15).
Factor out the common terms.
(a – 15) (a + 3)
Then, a2 – 12a – 45 = (a – 15) (a + 3).
(iii) The Given expression is a2 – 7a – 30.
By comparing the given expression a2 – 7a – 30 with the basic expression x^2 + ax + b.
Here, a = 1, b = -7, and c = -30.
The sum of two numbers is m + n = b = -7 = -10 + 3.
The product of two number is m * n = a * c = 1 * (-30) = -30 = -10 * 3
From the above two instructions, we can write the values of two numbers m and n as -10 and 3.
Then, a2 – 7a – 30 = a2 – 10a + 3a – 30.
= a (a – 10) + 3(a – 10).
Factor out the common terms.
(a – 10) (a + 3)
Then, a2 – 7a – 30 = (a – 10) (a + 3).
(iv) The Given expression is a2 – 5a – 24.
By comparing the given expression a2 – 5a – 24 with the basic expression x^2 + ax + b.
Here, a = 1, b = -5, and c = -24.
The sum of two numbers is m + n = b = -5 = -8 + 3.
The product of two number is m * n = a * c = 1 * (-24) = -24 = -8 * 3
From the above two instructions, we can write the values of two numbers m and n as -8 and 3.
Then, a2 – 5a – 24 = a2 – 8a + 3a – 24.
= a (a – 8) + 3(a – 8).
Factor out the common terms.
(a – 8) (a + 3)
Then, a2 – 5a – 24 = (a – 8) (a + 3).
(v) The Given expression is 3a2 + 10a + 8.
By comparing the given expression 3a2 + 10a + 8 with the basic expression ax^2 + bx + c.
Here, a = 3, b = 10, and c = 8.
The sum of two numbers is m + n = b = 10 = 6 + 4.
The product of two number is m * n = a * c = 3 * (8) = 24 = 6 * 4
From the above two instructions, we can write the values of two numbers m and n as 6 and 4.
Then, 3a2 + 10a + 8 = 3a2 + 6a + 4a + 8.
= 3a (a + 2) + 4(a + 2).
Factor out the common terms.
(a + 2) (3a + 4)
Then, 3a2 + 10a + 8 = (a + 2) (3a + 4).
(vi) The Given expression is 3a2 + 14a + 8.
By comparing the given expression 3a2 + 14a + 8 with the basic expression ax^2 + bx + b.
Here, a = 3, b = 14, and c = 8.
The sum of two numbers is m + n = b = 14 = 12 + 2.
The product of two number is m * n = a * c = 3 * (8) = 24 = 8 * 3
From the above two instructions, we can write the values of two numbers m and n as 8 and 3.
Then, 3a2 + 14a + 8 = 3a2 + 8a + 3a + 8.
= a (3a + 8) + 1 (3a + 8).
Factor out the common terms.
(3a + 8) (a + 1)
Then, 3a2 + 14a + 8 = (3a + 8) (a + 1).
(vii) The Given expression is 2a2 + a – 45.
By comparing the given expression 2a2 + a – 45 with the basic expression ax^2 + bx + b.
Here, a = 2, b = 1, and c = -45.
The sum of two numbers is m + n = b = 1 = 10 – 9.
The product of two number is m * n = a * c = 2 * (-45) = -90 = 10 * -9
From the above two instructions, we can write the values of two numbers m and n as 10 and -9.
Then, 2a2 + a – 45 = 2a2 + 10a – 9a – 45.
= 2a (a + 5) – 9 (a + 5).
Factor out the common terms.
(a + 5) (2a – 9)
Then, 2a2 + a – 45 = (a + 5) (2a – 9).
(viii) The Given expression is 6a2 + 11a – 10.
By comparing the given expression 6a2 + 11a – 10 with the basic expression ax^2 + bx + b.
Here, a = 6, b = 11, and c = -10.
The sum of two numbers is m + n = b = 11 = 15 – 4.
The product of two number is m * n = a * c = 6 * (-10) = -60 = 15 * -4
From the above two instructions, we can write the values of two numbers m and n as 15 and -4.
Then, 6a2 + 11a – 10 = 6a2 + 15a – 4a – 10.
= 3a (2a + 5) – 2 (2a + 5).
Factor out the common terms.
(2a + 5) (3a – 2)
Then, 6a2 + 11a – 10 = (2a + 5) (3a – 2).
(ix) The Given expression is 3a2 – 10a + 8.
By comparing the given expression 3a2 – 10a + 8 with the basic expression ax^2 + bx + b.
Here, a = 3, b = -10, and c = 8.
The sum of two numbers is m + n = b = -10 = -6 – 4.
The product of two number is m * n = a * c = 3 * (8) = 24 = -6 * -4
From the above two instructions, we can write the values of two numbers m and n as -6 and -4.
Then, 3a2 – 10a + 8 = 3a2 – 6a – 4a + 8.
= 3a (a – 2) – 4 (a – 2).
Factor out the common terms.
(a – 2) (3a – 4)
Then, 3a2 – 10a + 8 = (a – 2) (3a – 4).
(x) The Given expression is 2a2 – 17a – 30.
By comparing the given expression 2a2 – 17a – 30 with the basic expression ax^2 + bx + b.
Here, a = 2, b = -17, and c = -30.
The sum of two numbers is m + n = b = -17 = -20 + 3.
The product of two number is m * n = a * c = 2 * (-30) = -60 = -20 * 3
From the above two instructions, we can write the values of two numbers m and n as -20 and 3.
Then, 2a2 – 17a – 30 = 2a2 – 20a + 3a – 30.
= 2a (a – 10) + 3 (a – 10).
Factor out the common terms.
(a – 10) (2a + 3)
Then, 2a2 – 17a – 30 = (a – 10) (2a + 3).