Example 1 :
Factor :
2a2 + 9a + 10
Solution :
Since it is a quadratic expression and coefficient of a2 is 2, we have to multiply the constant 10 by 2.
10 ⋅ 2 = 20
Now we have to find two factors of 20 such that the product is 20 and sum is 9, which is the coefficient of x.
5 ⋅ 4 = 20 and 5 + 4 = 9
Then,
2a2 + 9a + 10 = 2a2 + 5a + 4a + 10
= a(2a + 5) + 2(2a + 5)
= (a + 2)(2a + 5)
So, the factors are (a + 2) and (2a + 5).
Example 2 :
Factor :
5x2 - 29xy - 42y2
Solution :
The product of 5 and -42 is -210. Now we have to find two factors of -210 such that the product is -210 and sum is -29, which is the coefficient of x.
-35 ⋅ 6 = -210 and -35 + 6 = -29
Then,
5x2 - 29xy - 42y2 = 5x2 - 35xy + 6xy - 42y2
= 5x(x - 7y) + 6y(x - 7y)
= (5x + 6y)(x - 7y)
So, the factors are (5x + 6y) and (x - 7y).
Example 3 :
Factor :
9 - 18x + 8x2
Solution :
8x2 - 18x + 9
The product of 8 and 9 is 72.
Now we have to find two factors of 72 such that the product is 72 and sum is -18, which is the coefficient of x.
-12 ⋅ (-6) = 72 and -12 - 6 = -18
8x2 - 18x + 9 = 8x2 - 12x - 6x + 9
= 4x(2x - 3) - 3(2x - 3)
= (4x - 3)(2x - 3)
So the factors are (4x - 3) and (2x - 3).
Example 4 :
Factor :
6x2 + 16xy + 8y2
Solution :
Factor out the greatest common factor.
6x2 + 16xy + 8y2 = 2(3x2 + 8xy + 4y2)
The product of 3 and 4 is 12.
Now we have to find two factors of 12 such that the product is 12 and sum is 8, which is the coefficient of x.
2 ⋅ 6 = 12 and 2 + 6 = 8
Then,
= 2(3x2 + 2xy + 6xy + 4y2)
= 2[x(3x + 2y) + 2y(3x + 2y)]
= 2(3x + 2y)(x + 2y)
So, the factors are 2, (3x + 2y) and (x + 2y).
Example 5 :
Factor :
12x2 + 36x2y + 27x2y2
Solution :
Factor out the greatest common factor.
12x2 + 36x2y + 27x2y2 = 3x2(4 + 12y + 9y2)
= 3x2(9y2 + 12y + 4)
The product of 9 and 4 is 36.
Now we have to find two factors of 36, such that the product is 36 and sum is 12, which is the coefficient of x.
6 ⋅ 6 = 36 and 6 + 6 = 12
Then,
= 3x2(9y2 + 6y + 6y + 4)
= 3x2[(3y(3y + 2) + 2(3y + 2)]
= 3x2(3y + 2)(3y + 2)
= 3x2(3y + 2)2
So, factors are 3, x2 and (3y + 2)2.
Example 6 :
Factor :
(a + b)2 + 9(a + b) + 18
Solution :
Let y = a + b.
(a + b)2 + 9(a + b) + 18 = y2 + 9y + 18
Now we have to find two factors of 18, such that the product is 18 and sum is 9, which is the coefficient of y.
3 ⋅ 6 = 18 and 3 + 6 = 9
Then,
= y2 + 3y + 6y + 18
= y(y + 3) + 6(y + 3)
= (y + 3)(y + 6)
Substitute (a + b) for y.
= (a + b + 3)(a + b + 6)
So, the factors are (a + b + 3) and (a + b + 6).
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