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1. Factorize the following expression:

(i) a2 + 9a + 20
(ii) m2 + 15m + 54
(iii) y2 + 3y – 4
(iv) n2 + 2n – 24
(v) x2 – 5x + 4
(vi) a2 – 15a + 14

Solution:

(i) 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).

(ii) The Given expression is m2 + 15m + 54.
By comparing the given expression m2 + 15m + 54 with the basic expression x^2 + ax + b.
Here, a = 1, b = 15, and c = 54.
The sum of two numbers is m + n = b = 15 = 6 + 9.
The product of two number is m * n = a * c = 1 * (54) = 54 = 6 * 9
From the above two instructions, we can write the values of two numbers m and n as 6 and 9.
Then, m2 + 15m + 54 = m2 + 6m + 9m + 54.
= m (m + 6) + 9(m + 6).
Factor out the common terms.
(m + 6) (m + 9)

Then, m2 + 15m + 54 = (m + 6) (m + 9).

(iii) The Given expression is y2 + 3y – 4.
By comparing the given expression y2 + 3y – 4 with the basic expression x^2 + ax + b.
Here, a = 1, b = 3, and c = -4.
The sum of two numbers is m + n = b = 3 = 4 – 1.
The product of two number is m * n = a * c = 1 * (-4) = -4 = 1 * -4
From the above two instructions, we can write the values of two numbers m and n as 1 and -4.
Then, y2 + 3y – 4 = y2 + 4y -y – 4.
= y (y + 4) – 1(y + 4).
Factor out the common terms.
(y + 4) (y – 1)

Then, y2 + 3y – 4 = (y + 4) (y – 1).

(iv) The Given expression is n2 + 2n – 24.
By comparing the given expression n2 + 2n – 24 with the basic expression x^2 + ax + b.
Here, a = 1, b = 2, and c = -24.
The sum of two numbers is m + n = b = 2 = 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, n2 + 2n – 24 = n2 + 6n – 4n – 24.
= n (n + 6) – 4(n + 6).
Factor out the common terms.
(n + 6) (n – 4)

Then, n2 + 2n – 24 = (n + 6) (n – 4).

(v) The Given expression is x2 – 5x + 4.
By comparing the given expression x2 – 5x + 4 with the basic expression x^2 + ax + b.
Here, a = 1, b = -5, and c = 4.
The sum of two numbers is m + n = b = -5 = -1 – 4.
The product of two number is m * n = a * c = 1 * (4) = 4 = -1 * -4
From the above two instructions, we can write the values of two numbers m and n as -1 and -4.
Then, x2 – 5x + 4 = x2 – x – 4x + 4.
= x (x – 1) – 4(x – 1).
Factor out the common terms.
(x – 1) (x – 4)

Then, x2 – 5x + 4 = (x – 1) (x – 4).

(vi) The Given expression is a2 – 15a + 14.
By comparing the given expression a2 – 15a + 14 with the basic expression x^2 + ax + b.
Here, a = 1, b = -15, and c = 14.
The sum of two numbers is m + n = b = -15 = -1 – 14.
The product of two number is m * n = a * c = 1 * (14) = 14 = -1 * -14
From the above two instructions, we can write the values of two numbers m and n as -1 and -14.
Then, a2 – 15a + 14 = a2 – a – 14a + 14.
= a (a – 1) – 14(a – 1).
Factor out the common terms.
(a – 1) (a – 14)

Then, a2 – 15a + 14 = (a – 1) (a – 14).

2. Resolve into factors

(i) a2 + 3a – 10
(ii) m2 – 18m – 63
(iii) a2 + 6a + 8
(iv) a2 + 12a + 32
(v) x2 – 8x + 15
(vi) m2 – 12m + 35

Solution:

(i) The Given expression is a2 + 3a – 10.
By comparing the given expression a2 + 3a – 10 with the basic expression x^2 + ax + b.
Here, a = 1, b = 3, and c = -10.
The sum of two numbers is m + n = b = 3 = 5 – 2.
The product of two number is m * n = a * c = 1 * (-10) = -10 = 5 * -2
From the above two instructions, we can write the values of two numbers m and n as 5 and -2.
Then, a2 + 3a – 10 = a2 + 5a – 2a – 10.
= a (a + 5) – 2(a + 5).
Factor out the common terms.
(a + 5) (a – 2)

Then, a2 + 3a – 10 = (a + 5) (a – 2).

(ii) The Given expression is m2 – 18m – 63.
By comparing the given expression m2 – 18m – 63 with the basic expression x^2 + ax + b.
Here, a = 1, b = -18, and c = -63.
The sum of two numbers is m + n = b = -18 = 3 – 21.
The product of two number is m * n = a * c = 1 * (-63) = -63 = 3 * -21
From the above two instructions, we can write the values of two numbers m and n as 3 and -21.
Then, m2 – 18m – 63 = m2 + 3m – 21m – 63.
= m (m + 3) – 21(m + 3).
Factor out the common terms.
(m + 3) (m – 21)

Then, m2 – 18m – 63 = (m + 3) (m – 21).

(iii) The Given expression is a2 + 6a + 8.
By comparing the given expression a2 + 6a + 8 with the basic expression x^2 + ax + b.
Here, a = 1, b = 6, and c = 8.
The sum of two numbers is m + n = b = 6 = 2 + 4.
The product of two number is m * n = a * c = 1 * (8) = 8 = 2 * 4
From the above two instructions, we can write the values of two numbers m and n as 2 and 4.
Then, a2 + 6a + 8 = a2 + 2a + 4a + 8.
= a (a + 2) + 4(a + 2).
Factor out the common terms.
(a + 2) (a + 4)

Then, a2 + 6a + 8 = (a + 2) (a + 4).

(iv) The Given expression is a2 + 12a + 32.
By comparing the given expression a2 + 12a + 32 with the basic expression x^2 + ax + b.
Here, a = 1, b = 12, and c = 32.
The sum of two numbers is m + n = b = 12 = 8 + 4.
The product of two number is m * n = a * c = 1 * (32) = 32 = 8 * 4
From the above two instructions, we can write the values of two numbers m and n as 8 and 4.
Then, a2 + 12a + 32 = a2 + 8a + 4a + 32.
= a (a + 8) + 4(a + 8).
Factor out the common terms.
(a + 8) (a + 4)

Then, a2 + 12a + 32 = (a + 8) (a + 4).

(v) The Given expression is x2 – 8x + 15.
By comparing the given expression x2 – 8x + 15 with the basic expression x^2 + ax + b.
Here, a = 1, b = -8, and c = 15.
The sum of two numbers is m + n = b = -8 = – 3 – 5.
The product of two number is m * n = a * c = 1 * (15) = 15 = -3 * -5
From the above two instructions, we can write the values of two numbers m and n as -3 and -5.
Then, x2 – 8x + 15 = x2 – 3x – 5x + 15.
= x (x – 3) – 5(x – 3).
Factor out the common terms.
(x – 3) (x – 5)

Then, x2 – 8x + 15 = (x – 3) (x – 5).

(vi) The Given expression is m2 – 12m + 35.
By comparing the given expression m2 – 12m + 35 with the basic expression x^2 + ax + b.
Here, a = 1, b = -12, and c = 35.
The sum of two numbers is m + n = b = -12 = – 5 – 7.
The product of two number is m * n = a * c = 1 * (35) = 35 = -5 * -7
From the above two instructions, we can write the values of two numbers m and n as -5 and -7.
Then, m2 – 12m + 35 = m2 – 5m – 7m + 35.
= m (m – 5) – 7(m – 5).
Factor out the common terms.
(m – 7) (m – 5)

Then, m2 – 12m + 35 = (m – 7) (m – 5).

3. Factor the middle term

(i) m2 – 4m – 12
(ii) a2 – 4a – 45
(iii) x2 + 15x + 56
(iv) p2 – 13p + 36
(v) q2 + 5q – 24
(vi) r2 + 17r – 84
(vii) a2 – 15a + 44
(viii) m2 – 5m – 24
(ix) x2 – 4x – 77
(x) a2 – 12a + 20

Solution:

(i) The Given expression is m2 – 4m – 12.
By comparing the given expression m2 – 4m – 12 with the basic expression x^2 + ax + b.
Here, a = 1, b = -4, and c = -12.
The sum of two numbers is m + n = b = -4 = 2 – 6.
The product of two number is m * n = a * c = 1 * (-12) = -12 = 2 * -6
From the above two instructions, we can write the values of two numbers m and n as 2 and -6.
Then, m2 – 4m – 12 = m2 – 2m + 6m – 12.
= m (m – 2) + 6(m – 2).
Factor out the common terms.
(m – 2) (m + 6)

Then, m2 – 4m – 12 = (m – 2) (m + 6).

(ii) The Given expression is a2 – 4a – 45.
By comparing the given expression a2 – 4a – 45 with the basic expression x^2 + ax + b.
Here, a = 1, b = -4, and c = -45.
The sum of two numbers is m + n = b = -4 = 5 – 9.
The product of two number is m * n = a * c = 1 * (-45) = -45 = 5 * -9
From the above two instructions, we can write the values of two numbers m and n as 5 and -9.
Then, a2 – 4a – 45 = a2 + 5a – 9a – 45.
= a (a + 5) – 9(a + 5).
Factor out the common terms.
(a + 5) (a – 9)

Then, a2 – 4a – 45 = (a + 5) (a – 9).

(iii) The Given expression is x2 + 15x + 56.
By comparing the given expression x2 + 15x + 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, x2 + 15x + 56 = x2 + 8x + 7x + 56.
= x (x + 8) + 7(x + 8).
Factor out the common terms.
(x + 8) (x + 7)

Then, x2 + 15x + 56 = (x + 8) (x + 7).

(iv) The Given expression is p2 – 13p + 36.
By comparing the given expression p2 – 13p + 36 with the basic expression x^2 + ax + b.
Here, a = 1, b = -13, and c = 36.
The sum of two numbers is m + n = b = -13 = -9 – 4.
The product of two number is m * n = a * c = 1 * (36) = 36 = -9 * -4
From the above two instructions, we can write the values of two numbers m and n as -9 and -4.
Then, p2 – 13p + 36 = p2 – 9p – 4p + 36.
= p (p – 9) – 4(p – 9).
Factor out the common terms.
(p – 9) (p – 4)

Then, p2 – 13p + 36 = (p – 9) (p – 4).

(v) The Given expression is q2 + 5q – 24.
By comparing the given expression q2 + 5q – 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 = -3 + 8.
The product of two number is m * n = a * c = 1 * (-24) = -24 = -3 * 8
From the above two instructions, we can write the values of two numbers m and n as -3 and 8.
Then, q2 + 5q – 24 = q2 – 3q + 8q – 24.
= q (q – 3) + 8(q – 3).
Factor out the common terms.
(q – 3) (q + 8)

Then, q2 + 5q – 24 = (q – 3) (q + 8).

(vi) The Given expression is r2 + 17r – 84.
By comparing the given expression r2 + 17r – 84 with the basic expression x^2 + ax + b.
Here, a = 1, b = 17, and c = -84.
The sum of two numbers is m + n = b = 17 = 21 – 4.
The product of two number is m * n = a * c = 1 * (-84) = -84 = 21 * -4
From the above two instructions, we can write the values of two numbers m and n as 21 and -4.
Then, r2 + 17r – 84 = r2 + 21r -4r – 84.
= r (r + 21) – 4(r + 21).
Factor out the common terms.
(r + 21) (r – 4)

Then, r2 + 17r – 84 = (r + 21) (r – 4).

(vii) The Given expression is a2 – 15a + 44.
By comparing the given expression a2 – 15a + 44 with the basic expression x^2 + ax + b.
Here, a = 1, b = -15, and c = 44.
The sum of two numbers is m + n = b = -15 = -11 – 4.
The product of two number is m * n = a * c = 1 * (44) = 44 = -11 * -4
From the above two instructions, we can write the values of two numbers m and n as -11 and -4.
Then, a2 – 15a + 44 = a2 – 11a – 4a + 44.
= a (a – 11) – 4(a – 11).
Factor out the common terms.
(a – 11) (a – 4)

Then, a2 – 15a + 44 = (a – 11) (a – 4).

(viii) The Given expression is m2 – 5m – 24.
By comparing the given expression m2 – 5m – 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 = 3 – 8.
The product of two number is m * n = a * c = 1 * (-24) = -24 = 3 * -8
From the above two instructions, we can write the values of two numbers m and n as 3 and -8.
Then, m2 – 5m – 24 = m2 + 3m – 8m – 24.
= m (m + 3) – 8(m – 3).
Factor out the common terms.
(m – 3) (m – 8)

Then, m2 – 5m – 24 = (m – 3) (m – 8).

(ix) The Given expression is x2 – 4x – 77.
By comparing the given expression x2 – 4x – 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, x2 – 4x – 77 = x2 – 11x +7x – 77.
= x (x – 11) + 7(x – 11).
Factor out the common terms.
(x – 11) (x + 7)

Then, x2 – 4x – 77 = (x – 11) (x + 7).

(x) The Given expression is a2 – 12a + 20.
By comparing the given expression a2 – 12a + 20 with the basic expression x^2 + ax + b.
Here, a = 1, b = -12, and c = 20.
The sum of two numbers is m + n = b = -12 = -10 – 2.
The product of two number is m * n = a * c = 1 * (20) = 20 = -10 * -2
From the above two instructions, we can write the values of two numbers m and n as -10 and -2.
Then, a2 – 12a + 20 = a2 – 10a – 2a + 20.
= a (a – 10) – 2(a – 10).
Factor out the common terms.
(a – 10) (a – 2)

Then, a2 – 12a + 20 = (a – 10) (a – 2).