The quadratic expression which is in the form of
ax^{2}+bx+c
can be factored using the steps given below.
Step 1 :
Check whether the coefficient of x^{2 }is 1 or not equal to 1. If it is 1, you can take the constant alone. If it is not equal to 1, multiply the constant by the leading coefficient and decompose it into two term.
Step 2 :
Decompose (ac) as two terms. So, the product is equal to ac and when we simplify those two terms, we should get the middle term.
Step 3 :
Rewrite the middle term in terms of factors. Using grouping, find the factors.
Example 1 :
5x^{2} – 8x + 3
Solution :
Step 1 :
Product of factors = 15
Sum of factors = -8
Step 2 :
1 · 15 = 15 and 1 + 15 = 16 ≠ -8
(-5) · (-3) = 15 and (-5) + (-3) = -8
So, the factors are -5 and -3.
Step 3 :
= 5x^{2} - 5x - 3x + 3
By grouping,
= (5x^{2} – 5x) + (-3x + 3)
Step 4 :
By taking the common factor, we get
= 5x(x - 1) - 3(x – 1)
= (x – 1)(5x - 3)
Example 2 :
11x^{2} – 9x - 2
Solution :
Step 1 :
Product of factors = -22
Sum of factors = -9
Step 2 :
(-1) · 22 = -22 and (-1) + 22 = 21 ≠ -9
(-11) · 2 = -22 and (-11) + 2 = -9
So, the factors are -11 and 2.
Step 3 :
= 11x^{2} - 11x + 2x - 2
By grouping,
= (11x^{2} – 11x) + (2x – 2)
Step 4 :
By taking the common factor, we get
= 11x(x - 1) + 2(x – 1)
= (x - 1)(11x + 2)
Example 3 :
3x^{2} – 7x - 6
Solution :
Step 1 :
Product of factors = -18
Sum of factors = -7
Step 2 :
(-6) · 3 = -18 and (-6) + 3 = -3 ≠ -7
(-9) · 2 = -18 and (-9) + 2 = -7
So, the factors are -9 and 2.
Step 3 :
= 3x^{2} - 9x + 2x – 6
By grouping,
= (3x^{2} - 9x) + (2x - 6)
Step 4 :
By taking the common factor, we get
= 3x(x - 3) + 2(x – 3)
= (x - 3)(3x + 2)
Example 4 :
2x^{2} – 3x - 9
Solution :
Step 1 :
Product of factors = -18
Sum of factors = -3
Step 2 :
(-9) · 2 = -18 and (-9) + 2 = -7 ≠ -3
(-6) · 3 = -18 and (-6) + 3 = -3
So, the factors are -6 and 3.
Step 3 :
= 2x^{2} - 6x + 3x – 9
By grouping,
= (2x^{2} - 6x) + (3x - 9)
Step 4 :
By taking the common factor, we get
= 2x(x - 3) + 3(x - 3)
= (x - 3)(2x + 3)
Example 5 :
3x^{2} + 10x - 8
Solution :
Product of factors = -24
Sum of factors = 10
(-8) · 3 = -24 and (-8) + 3 = -5 ≠ 10
12 · (-2) = -24 and 12 + (-2) = 10
= 3x^{2} + 12x - 2x – 8
By grouping,
= (3x^{2} + 12x) + (-2x - 8)
By taking the common factor, we get
= 3x(x + 4) - 2(x + 4)
= (x + 4)(3x – 2)
Example 6 :
2x^{2} + 9x – 18
Solution :
Product of factors = -36
Sum of factors = 9
(-9) · 4 = -36 and (-9) + 4 = -5 ≠ 9
12 · (-3) = -36 and 12 + (-3) = 9
So, the factors are 12 and -3.
= 2x^{2} + 12x - 3x – 18
By grouping,
= (2x^{2} + 12x) + (-3x – 18)
By taking the common factor, we get
= 2x(x + 6) - 3(x + 6)
= (2x – 3)(x + 6)
Example 7 :
2x^{2} + 11x – 21
Solution :
Product of factors = -42
Sum of factors = 11
(-21) · 2 = -42 and (-21) + 2 = -19 ≠ 11
14 · (-3) = -42 and 14 + (-3) = 11
So, the factors are 14 and -3.
= 2x^{2} + 14x - 3x – 21
By grouping,
= (2x^{2} + 14x) + (-3x - 21)
By taking the common factor, we get
= 2x(x + 7) - 3(x + 7)
= (2x - 3)(x + 7)
Example 8 :
15x^{2} + x - 2
Solution :
Product of factors = -30
Sum of factors = 1
(-10) · 3 = -30 and (-10) + 3 = -7 ≠ 1
(-5) · 6 = -30 and (-5) + 6 = 1
So, the factors are -5 and 6.
= 15x^{2} - 5x + 6x – 2
By grouping,
= (15x^{2} - 5x) + (6x – 2)
By taking the common factor, we get
= 5x(3x - 1) + 2(3x - 1)
= (5x + 2)(3x - 1)
Example 9 :
21x^{2} – 62x - 3
Solution :
Product of factors = -63
Sum of factors = -62
(-9) · 7 = -63 and (-9) + 7 = -2 ≠ -62
1 · (-63) = -63 and 1 + (-63) = -62
So, the factors are 1 and -63.
= 21x^{2} + x - 63x – 3
By grouping,
= (21x^{2} + x) + (-63x - 3)
By taking the common factor, we get
= x(21x + 1) - 3(21x + 1)
= (x - 3)(21x + 1)
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