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1. Factorize the quadratic trinomials

(i) a^{2} + 5a + 6

(ii) a^{2} + 10a + 24

(iii) a^{2} + 12a + 27

(iv) a^{2} + 15a + 56

(v) a^{2} + 19a + 60

(vi) a^{2} + 13a + 40

(vii) a^{2} – 10a + 24

(viii) a^{2} – 23a + 42

(ix) a^{2} – 17a + 16

(x) a^{2} – 21a + 90

## Solution:

(i) The Given expression is a^{2} + 5a + 6.

By comparing the given expression a^{2} + 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, a^{2} + 5a + 6 = a^{2} + 3a + 2a + 6.

= a (a+ 3) + 2(a + 3).

Factor out the common terms.

(a + 3) (a + 2)

Then, a^{2} + 5a + 6 = (a + 3) (a + 2).

(ii) The Given expression is a^{2} + 10a + 24.

By comparing the given expression a^{2} + 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, a^{2} + 10a + 24 = a^{2} + 6a + 4a + 24.

= a (a+ 6) + 4(a + 6).

Factor out the common terms.

(a + 6) (a + 4)

Then, a^{2} + 10a + 24 = (a + 6) (a + 4).

(iii) The Given expression is a^{2} + 12a + 27.

By comparing the given expression a^{2} + 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, a^{2} + 12a + 27 = a^{2} + 9a + 3a + 27.

= a (a+ 9) + 3(a + 9).

Factor out the common terms.

(a + 9) (a + 3)

Then, a^{2} + 12a + 27 = (a + 9) (a + 3).

(iv) The Given expression is a^{2} + 15a + 56.

By comparing the given expression a^{2} + 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, a^{2} + 15a + 56 = a^{2} + 8a + 7a + 56.

= a (a+ 8) + 7(a + 8).

Factor out the common terms.

(a + 8) (a + 7)

Then, a^{2} + 15a + 56 = (a + 8) (a + 7).

(v) The Given expression is a^{2} + 19a + 60.

By comparing the given expression a^{2} + 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, a^{2} + 19a + 60 = a^{2} + 15a + 4a + 60.

= a (a+ 15) + 4(a + 15).

Factor out the common terms.

(a + 15) (a + 4)

Then, a^{2} + 19a + 60 = (a + 15) (a + 4).

(vi) The Given expression is a^{2} + 13a + 40.

By comparing the given expression a^{2} + 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, a^{2} + 13a + 40 = a^{2} + 8a + 5a + 40.

= a (a + 8) + 5(a + 8).

Factor out the common terms.

(a + 8) (a + 5)

Then, a^{2} + 13a + 40 = (a + 8) (a + 5).

(vii) The Given expression is a^{2} – 10a + 24.

By comparing the given expression a^{2} – 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, a^{2} – 10a + 24 = a^{2} – 6a – 4a + 24.

= a (a – 6) – 4(a – 6).

Factor out the common terms.

(a – 6) (a – 4)

Then, a^{2} – 10a + 24 = (a – 6) (a – 4).

(viii) The Given expression is a^{2} – 23a + 42.

By comparing the given expression a^{2} – 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, a^{2} – 23a + 42 = a^{2} – 21a – 2a + 42.

= a (a – 21) – 2(a – 21).

Factor out the common terms.

(a – 21) (a – 2)

Then, a^{2} – 23a + 42 = (a – 21) (a – 2).

(ix) The Given expression is a^{2} – 17a + 16.

By comparing the given expression a^{2} – 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, a^{2} – 17a + 16 = a^{2} – 16a -a + 16.

= a (a – 16) – 1(a – 16).

Factor out the common terms.

(a – 16) (a – 1)

Then, a^{2} – 17a + 16 = (a – 16) (a – 1).

(x) The Given expression is a^{2} – 21a + 90.

By comparing the given expression a^{2} – 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, a^{2} – 21a + 90 = a^{2} – 15a – 6a + 90.

= a (a – 15) – 6(a – 15).

Factor out the common terms.

(a – 15) (a – 6)

Then, a^{2} – 21a + 90 = (a – 15) (a – 6).

2. Factorize the expressions completely

(i) a^{2} – 22a + 117

(ii) a^{2} – 9a + 20

(iii) a^{2} + a – 132

(iv) a^{2} + 5a – 104

(v) b^{2} + 7b – 144

(vi) c^{2} + 19c – 150

(vii) b^{2} + b – 72

(viii) a^{2} + 6a – 91

(ix) a^{2} – 4a -77

(x) a^{2} – 6a – 135

## Solution:

(i) The Given expression is a^{2} – 22a + 117.

By comparing the given expression a^{2} – 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, a^{2} – 22a + 117 = a^{2} – 13a – 9a + 117.

= a (a – 13) – 9(a – 13).

Factor out the common terms.

(a – 13) (a – 9)

Then, a^{2} – 22a + 117 = (a – 13) (a – 9).

(ii) The Given expression is a^{2} – 9a + 20.

By comparing the given expression a^{2} – 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, a^{2} – 9a + 20 = a^{2} – 5a -4a + 20.

= a (a – 5) – 4(a – 5).

Factor out the common terms.

(a – 5) (a – 4)

Then, a^{2} – 9a + 20 = (a – 5) (a – 4).

(iii) The Given expression is a^{2} + a – 132.

By comparing the given expression a^{2} + 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, a^{2} + a – 132 = a^{2} + 12a – 11a – 132.

= a (a + 12) – 11(a + 12).

Factor out the common terms.

(a + 12) (a – 11)

Then, a^{2} + a – 132 = (a + 12) (a – 11).

(iv) The Given expression is a^{2} + 5a – 104.

By comparing the given expression a^{2} + 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, a^{2} + 5a – 104 = a^{2} + 13a – 8a – 104.

= a (a + 13) – 8(a + 13).

Factor out the common terms.

(a + 13) (a – 8)

Then, a^{2} + 5a – 104 = (a + 13) (a – 8).

(v) The Given expression is b^{2} + 7b – 144.

By comparing the given expression b^{2} + 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, b^{2} + 7b – 144 = b^{2} + 16b – 9b – 144.

= b (b + 16) – 9(b + 16).

Factor out the common terms.

(b + 16) (b – 9)

Then, b^{2} + 7b – 144 = (b + 16) (b – 9).

(vi) The Given expression is c^{2} + 19c – 150.

By comparing the given expression c^{2} + 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, c^{2} + 19c – 150 = c^{2} + 25c – 6c – 150.

= c (c + 25) – 6(c + 25).

Factor out the common terms.

(c + 25) (c – 6)

Then, c^{2} + 19c – 150 = (c + 25) (c – 6).

(vii) The Given expression is b^{2} + b – 72.

By comparing the given expression b^{2} + 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, b^{2} + b – 72 = b^{2} + 9b – 8b – 72.

= b (b + 9) – 8(b + 9).

Factor out the common terms.

(b + 9) (b – 8)

Then, b^{2} + b – 72 = (b + 9) (b – 8).

(viii) The Given expression is a^{2} + 6a – 91.

By comparing the given expression a^{2} + 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, a^{2} + 6a – 91 = a^{2} + 13a – 7a – 91.

= a (a + 13) – 7(a + 13).

Factor out the common terms.

(a + 13) (a – 7)

Then, a^{2} + 6a – 91 = (a + 13) (a – 7).

(ix) The Given expression is a^{2} – 4a -77.

By comparing the given expression a^{2} – 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, a^{2} – 4a -77 = a^{2} – 11a + 7a -77.

= a (a – 11) + 7(a – 11).

Factor out the common terms.

(a – 11) (a + 7)

Then, a^{2} – 4a -77 = (a – 11) (a + 7).

(x) The Given expression is a^{2} – 6a – 135.

By comparing the given expression a^{2} – 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, a^{2} – 6a – 135 = a^{2} – 15a + 9a – 135.

= a (a – 15) + 9(a – 15).

Factor out the common terms.

(a – 15) (a + 9)

Then, a^{2} – 6a – 135 = (a – 15) (a + 9).

3. Factor by splitting the middle term

(i) a^{2} – 11a – 42

(ii) a^{2} – 12a – 45

(iii) a^{2} – 7a – 30

(iv) a^{2} – 5a – 24

(v) 3a^{2} + 10a + 8

(vi) 3a^{2} + 14a + 8

(vii) 2a^{2} + a – 45

(viii) 6a^{2} + 11a – 10

(ix) 3a^{2} – 10a + 8

(x) 2a^{2} – 17a – 30

## Solution:

(i) The Given expression is a^{2} – 11a – 42.

By comparing the given expression a^{2} – 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, a^{2} – 11a – 42 = a^{2} – 14a + 3a- 42.

= a (a – 14) + 3(a – 14).

Factor out the common terms.

(a – 14) (a + 3)

Then, a^{2} – 11a – 42 = (a – 14) (a + 3).

(ii) The Given expression is a^{2} – 12a – 45.

By comparing the given expression a^{2} – 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, a^{2} – 12a – 45 = a^{2} – 15a + 3a- 45.

= a (a – 15) + 3(a – 15).

Factor out the common terms.

(a – 15) (a + 3)

Then, a^{2} – 12a – 45 = (a – 15) (a + 3).

(iii) The Given expression is a^{2} – 7a – 30.

By comparing the given expression a^{2} – 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, a^{2} – 7a – 30 = a^{2} – 10a + 3a – 30.

= a (a – 10) + 3(a – 10).

Factor out the common terms.

(a – 10) (a + 3)

Then, a^{2} – 7a – 30 = (a – 10) (a + 3).

(iv) The Given expression is a^{2} – 5a – 24.

By comparing the given expression a^{2} – 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, a^{2} – 5a – 24 = a^{2} – 8a + 3a – 24.

= a (a – 8) + 3(a – 8).

Factor out the common terms.

(a – 8) (a + 3)

Then, a^{2} – 5a – 24 = (a – 8) (a + 3).

(v) The Given expression is 3a^{2} + 10a + 8.

By comparing the given expression 3a^{2} + 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, 3a^{2} + 10a + 8 = 3a^{2} + 6a + 4a + 8.

= 3a (a + 2) + 4(a + 2).

Factor out the common terms.

(a + 2) (3a + 4)

Then, 3a^{2} + 10a + 8 = (a + 2) (3a + 4).

(vi) The Given expression is 3a^{2} + 14a + 8.

By comparing the given expression 3a^{2} + 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, 3a^{2} + 14a + 8 = 3a^{2} + 8a + 3a + 8.

= a (3a + 8) + 1 (3a + 8).

Factor out the common terms.

(3a + 8) (a + 1)

Then, 3a^{2} + 14a + 8 = (3a + 8) (a + 1).

(vii) The Given expression is 2a^{2} + a – 45.

By comparing the given expression 2a^{2} + 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, 2a^{2} + a – 45 = 2a^{2} + 10a – 9a – 45.

= 2a (a + 5) – 9 (a + 5).

Factor out the common terms.

(a + 5) (2a – 9)

Then, 2a^{2} + a – 45 = (a + 5) (2a – 9).

(viii) The Given expression is 6a^{2} + 11a – 10.

By comparing the given expression 6a^{2} + 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, 6a^{2} + 11a – 10 = 6a^{2} + 15a – 4a – 10.

= 3a (2a + 5) – 2 (2a + 5).

Factor out the common terms.

(2a + 5) (3a – 2)

Then, 6a^{2} + 11a – 10 = (2a + 5) (3a – 2).

(ix) The Given expression is 3a^{2} – 10a + 8.

By comparing the given expression 3a^{2} – 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, 3a^{2} – 10a + 8 = 3a^{2} – 6a – 4a + 8.

= 3a (a – 2) – 4 (a – 2).

Factor out the common terms.

(a – 2) (3a – 4)

Then, 3a^{2} – 10a + 8 = (a – 2) (3a – 4).

(x) The Given expression is 2a^{2} – 17a – 30.

By comparing the given expression 2a^{2} – 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, 2a^{2} – 17a – 30 = 2a^{2} – 20a + 3a – 30.

= 2a (a – 10) + 3 (a – 10).

Factor out the common terms.

(a – 10) (2a + 3)

Then, 2a^{2} – 17a – 30 = (a – 10) (2a + 3).