For a clear understanding of the factorization trinomials by substitution, take the help of our worksheets. Follow the questions on Worksheet on Factoring Trinomials by Substitution for reference and grade up your skills level on the concept. Clear all your doubts on the factorization concept by following the below-solved examples of factoring trinomials by substitution.
We know the factorization process for x^2 + ax + b or ax^2 + bx + c. But, for the trinomial expression, we need to substitute the terms in the place of common factors. So that, we will get the expression in the form of x^2 + ax + b or ax^2 + bx + c. Check Factorization Worksheets to understand the complete factorization concept.
Solved Examples of Factoring Trinomials by Substitution
1. Factor the following trinomials using the substitution method
(i) 4(x – y)^2 – 14(x – y) – 8.
(ii) 3(3a + 2)^2 + 5(3a + 2) – 2.
(iii) 2(x + 2y)^2 + (x + 2y) – 1.
(iv) (x + y)^2 – (x + y) – 6.
(v) (a2 – 3a)^2 – 38(a^2 -3a) – 80.
(vi) 6(a – b)^2 – a + b – 15.
(vii) (x^2 – 3y^2)^2 – 16(x^2 – 3y^2) + 63.
(viii) (x^2 + 2x)^2 – 22(x^2 + 2x) + 72.
(ix) (a^2 – 8a)^2 – 29(a^2 – 8a) + 180.
(x) (p + q)^2 – 8p – 8q + 7.
Solution:
(i) The given expression is 4(x – y)^2 – 14(x – y) – 8.
Replace the (x – y) with p. That is 4p^2 – 14p – 8.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 4, b = – 14, c = – 8.
a*c = 4 * ( – 8) = – 32 and b = – 14.
So, – 32 = – 16 * 2 and – 14 = – 16 + 2.
Then, 4p^2 – 16 p + 2p – 8 = 4p(p – 4) + 2(p – 4).
Factor out the common terms from the above expression. That is,
(p – 4) (4p + 2).
Now, replace the term p with the (x – y). That is,
(x – y – 4) (4(x – y) + 2) = (x – y – 4) 2(2x – 2y + 1).
Then, 4(x – y)^2 – 14(x – y) – 8 is equal to 2(x – y – 4) (2x – 2y + 1).
(ii) The given expression is 3(3a + 2)^2 + 5(3a + 2) – 2.
Replace the (3a + 2) with p. That is,
3p^2 + 5p – 2.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 3, b = 5, c = – 2.
a*c = 3 * ( – 2) = – 6 and b = 5.
So, – 6 = 6 * (- 1) and 5= 6 – 1.
Then, 3p^2 + 5p – 2 = 3p^2 + 6p – p – 2.
= 3p(p + 2) – (p + 2).
Factor out the common terms from the above expression. That is,
(3p – 1)(p + 2).
Now, replace the term p with the (3a + 2). That is,
(3(3a + 2) – 1) (3a + 2 + 2) = (9a + 6 – 1) (3a + 4)
= (9a + 5) (3a + 4)
Then, 3(3a + 2)^2 + 5(3a + 2) – 2 is equal to (9a + 5) (3a + 4).
(iii) The given expression is 2(x + 2y)^2 + (x + 2y) – 1.
Replace the (x + 2y) with p. That is,
2p^2 + p – 1.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 2, b = 1, c = – 1.
a*c = 2 * ( – 1) = – 2 and b = 1.
So, – 2 = 2 * ( – 1) and 1 = 2 – 1.
Then, 2p^2 + p – 1 = 2p^2 + 2p – p – 1.
= 2p(p + 1) – (p + 1).
Factor out the common terms from the above expression. That is,
(2p – 1) (p + 1).
Now, replace the term p with the (x + 2y). That is,
(2p – 1) (p + 1) = (2(x + 2y) – 1) (x + 2y + 1).
= (2x + 4y – 1) (x + 2y + 1).
Then, 2(x + 2y)^2 + (x + 2y) – 1 is equal to (2x + 4y – 1) (x + 2y + 1).
(iv) The given expression is (x + y)^2 – (x + y) – 6.
Replace the (x + y) with p. That is,
p^2 – p – 6.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = – 1, c = – 6.
a*c = 1 * ( – 6) = – 6 and b = – 1.
So, – 6 = – 3 * 2 and – 1 = – 3 + 2.
Then, p^2 – p – 6 = p^2 – 3p + 2p – 6.
= p(p – 3) + 2(p – 3).
Factor out the common terms from the above expression. That is,
(p – 3) (p + 2).
Now, replace the term p with the (x + y). That is,
(p – 3) (p + 2) = (x + y – 3) (x + y + 2).
Then, (x + y)^2 – (x + y) – 6 is equal to (x + y – 3) (x + y + 2).
(v) The given expression is (a^2 – 3a)^2 – 38(a^2 -3a) – 80.
Replace the (a^2 – 3a) with p. That is,
p^2 – 38p – 80.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = – 38, c = – 80.
a*c = 1 * ( – 80) = – 80 and b = – 38.
So, – 80 = – 40 * 2 and – 38 = – 40 + 2.
Then, p^2 – 38p – 80 = p^2 – 40p + 2p – 80.
= p(p – 40) + 2(p – 40).
Factor out the common terms from the above expression. That is,
(p – 40) (p + 2).
Now, replace the term p with the (a^2 – 3a). That is,
(p – 40) (p + 2) = (a^2 – 3a – 40) ( a^2 – 3a + 2).
Then, (a2 – 3a)^2 – 38(a^2 -3a) – 80 is equal to (a^2 – 3a – 40) ( a^2 – 3a + 2).
(vi) The given expression is 6(a – b)^2 – a + b – 15.
We can write it as 6(a – b)^2 – (a – b) – 15.
Replace the (a – b) with p. That is,
6p^2 – p – 15.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 6, b = – 1, c = – 15.
a*c = 6 * ( – 15) = – 90 and b = – 1.
So, – 90 = – 10 * 9 and – 1 = – 10 + 9.
Then, 6p^2 – p – 15 = 6p^2 – 10p + 9p – 15.
= 2p(3p – 5) + 3(3p – 5).
Factor out the common terms from the above expression. That is,
(3p – 5) (2p + 3).
Now, replace the term p with the (a – b). That is,
(3p – 5) (2p + 3) = (3(a – b) – 5) (2(a – b) + 3).
Then, 6(a – b)^2 – a + b – 15 is equal to (3(a – b) – 5) (2(a – b) + 3).
(vii) The given expression is (x^2 – 3y^2)^2 – 16(x^2 – 3y^2) + 63.
Replace the (x^2 – 3y^2) with p. That is,
p^2 – 16p + 63.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = – 16, c = 63.
a*c = 1 * 63 = 63 and b = – 16.
So, 63 = – 7 * ( – 9) and – 16 = – 7 – 9.
Then, p^2 – 16p + 63 = p^2 – 9p – 7p + 63.
= p(p – 9) – 7(p – 9).
Factor out the common terms from the above expression. That is,
(p – 9) (p – 7).
Now, replace the term p with the (x^2 – 3y^2). That is,
(p – 9) (p – 7) = (x^2 – 3y^2 – 9) (x^2 – 3y^2 – 7).
Then, (x^2 – 3y^2)^2 – 16(x^2 – 3y^2) + 63 is equal to (x^2 – 3y^2 – 9) (x^2 – 3y^2 – 7).
(viii) The given expression is (x^2 + 2x)^2 – 22(x^2 + 2x) + 72.
Replace the (x^2 + 2x) with p. That is,
p^2 – 22p + 72.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = – 22, c = 72.
a*c = 1 * 72 = 72 and b = – 22.
So, 72 = – 18 * ( – 4) and – 22 = – 18 + ( – 4).
Then, p^2 – 22p + 72 = p^2 – 18p – 4p + 72.
= p(p – 18) – 4(p – 18).
Factor out the common terms from the above expression. That is,
(p – 18) (p – 4).
Now, replace the term p with the (x^2 + 2x). That is,
(p – 18) (p – 4) = (x^2 + 2x – 18) (x^2 + 2x – 4).
Then, (x^2 + 2x)^2 – 22(x^2 + 2x) + 72 is equal to (x^2 + 2x – 18) (x^2 + 2x – 4).
(ix) The given expression is (a^2 – 8a)^2 – 29(a^2 – 8a) + 180.
Replace the (a^2 – 8a) with p. That is,
p^2 – 29p + 180.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = – 29, c = 180.
a*c = 1 * 180 = 180 and b = – 29.
So, 180 = – 20 * (- 9) and – 29 = – 20 – 9.
Then, p^2 – 29p + 180 = p^2 – 9p – 20p + 180.
= p(p – 9) – 20(p – 9).
Factor out the common terms from the above expression. That is,
(p – 9) (p – 20).
Now, replace the term p with the (a^2 – 8a). That is,
(p – 9) (p – 20) = (a^2 – 8a – 9) (a^2 – 8a – 20).
Then, (a^2 – 8a)^2 – 29(a^2 – 8a) + 180 is equal to (a^2 – 8a – 9) (a^2 – 8a – 20).
(x) (p + q)2 – 8p – 8q + 7.
Solution: The given expression is
(p + q)2 – 8p – 8q + 7.
We can write it as (p + q)^2 – 8(p + q) + 7.
Replace the (p + q) with x. That is,
x^2 – 8x + 7.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = – 8, c = 7.
a*c = – 1 * (-7) = 7 and b = – 8.
So, 7 = – 7 * – 1and – 8 = – 7 – 1.
Then, x^2 – 8x + 7 = x^2 – 7x – x + 7.
= x(x – 7) – (x – 7).
Factor out the common terms from the above expression. That is,
(x – 7) (x – 1).
Now, replace the term x with the (p + q). That is,
(x – 7) (x – 1) = (p + q – 7) (p + q – 1).
Then, (p + q)2 – 8p – 8q + 7is equal to (p + q – 7) (p + q – 1).
2. Factor trinomials using substitution
(i) (a – 2b)^2 + 7(a – 2b) – 18
(ii) 3(x – y)^2 – (x – y) – 44
(iii) (5x – 3y)^2 + 8(5x – 3y) + 16
(iv) (a – 4b)^2 – 10(a – 4b) + 25
(v) (3x – 4)^2 – 4(3x – 4) – 12
(vi) (7x – 1)^2 + 12(7x – 1) – 45
Solution:
(i) The given expression is (a – 2b)^2 + 7(a – 2b) – 18.
Replace the (a – 2b) with x. That is,
x^2 + 7x – 18.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = 7, c = – 18.
a*c = 1 * (- 18) = – 18 and b = 7.
So, – 18 = 9 * ( -2) and 7 = 9 + ( – 2).
Then, x^2 + 7x – 18 = x^2 – 2x + 9x – 18.
= x(x – 2) + 9(x – 2).
Factor out the common terms from the above expression. That is,
(x – 2) (x + 9).
Now, replace the term x with the (a – 2b). That is,
(x – 2) (x + 9) = (a – 2b – 2) (a – 2b+ 9).
Then, (a – 2b)^2 + 7(a – 2b) – 18 is equal to (a – 2b – 2) (a – 2b + 9).
(ii) The given expression is 3(x – y)^2 – (x – y) – 44.
Replace the (x – y) with p. That is,
3p^2 – p – 44.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 3, b = – 1, c = – 44.
a*c = 3 * (- 44) = – 132 and b = – 1.
So, – 132 = – 12 * 11and – 1 = – 12 + 11.
Then, 3p^2 – p – 44 = 3p^2 – 12p + 11p – 44.
= 3p(p – 4) + 11(p – 4)
Factor out the common terms from the above expression. That is,
(p – 4) (3p + 11).
Now, replace the term p with the (x – y). That is,
(p – 4) (3p + 11) = (x – y – 4) (3(x – y) + 11).
Then, 3(x – y)^2 – (x – y) – 44 is equal to (x – y – 4) (3(x – y) + 11).
(iii) The given expression is (5x – 3y)^2 + 8(5x – 3y) + 16.
Replace the (5x – 3y) with p. That is,
p^2 + 8p + 16.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = 8, c = 16.
a*c = 1 * 16 = 16 and b = 8.
So, 16 = 4 * 4and 8 = 4 + 4.
Then, p^2 + 8p + 16 = p^2 + 4p + 4p + 16.
= p(p + 4) + 4(p + 4).
Factor out the common terms from the above expression. That is,
(p + 4) (p + 4).
Now, replace the term p with the (5x – 3y). That is,
(p +4) (p + 4) = (5x – 3y + 4) (5x – 3y + 4).
Then, (5x – 3y)^2 + 8(5x – 3y) + 16 is equal to (5x – 3y + 4) (5x – 3y + 4).
(iv) The given expression is (a – 4b)^2 – 10(a – 4b) + 25.
Replace the (a – 4b) with p. That is,
p^2 – 10p + 25.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = – 10, c = 25.
a*c = 1 * 25 = 25 and b = – 10.
So, 25 = – 5 * ( – 5) and – 10 = – 5 + ( – 5).
Then, p^2 – 10p + 25 = p^2 – 5p – 5p + 25.
= p(p – 5) – 5(p – 5).
Factor out the common terms from the above expression. That is,
(p – 5) (p – 5).
Now, replace the term p with the (a – 4b). That is,
(p – 5) (p – 5) = (a – 4b – 5) (a – 4b – 5).
Then, (a – 4b)^2 – 10(a – 4b) + 25 is equal to (a – 4b – 5) (a – 4b – 5).
(v) The given expression is (3x – 4)^2 – 4(3x – 4) – 12.
Replace the (3x – 4) with p. That is,
p^2 – 4p – 12.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = – 4, c = – 12.
a*c = 1 * ( – 12) = – 12 and b = – 4.
So, – 12 = – 6 * 2 and – 4 = – 6 + 2.
Then, p^2 – 4p – 12 = p^2 – 6p + 2p – 12.
= p(p – 6) + 2(p – 6).
Factor out the common terms from the above expression. That is,
(p – 6) (p + 2).
Now, replace the term p with the (3x – 4). That is,
(p – 6) (p + 2) = (3x – 4 – 6) (3x – 4 + 2).
= (3x – 10) (3x- 2).
Then, (3x – 4)^2 – 4(3x – 4) – 12 is equal to (3x – 10) (3x- 2).
(vi) The given expression is (p + q)^2 – 8p – 8q + 7.
Replace the (7x – 1) with p. That is,
p^2 + 12p – 45.
The above expression matches with the basic expression ax^2 + bx + c.
Here, a = 1, b = 12, c = – 45.
a*c = 1 * ( – 45) = – 45 and b = 12.
So, – 45 = 15 * ( – 3) and 12 = 15 + ( – 3).
Then, p^2 + 12p – 45 = p^2 + 15p – 3p – 45.
= p(p + 15) – 3(p + 15).
Factor out the common terms from the above expression. That is,
(p + 15) (p – 3).
Now, replace the term p with the (7x – 1). That is,
(p + 15) (p – 3) = (7x – 1 + 15) (7x – 1 – 3).
= (7x + 14) (7x – 4).
Then, (p + q)^2 – 8p – 8q + 7 is equal to (7x + 14) (7x – 4).