Example 1 :
x^{2} + 3x – 54 = 0
Solution :
Product of factors = -54
Sum of factors = 3
Finding the factors of product and sum.
(-27) · 2 = -54 and (-27) + 2 ≠ -25
9 · (-6) = -54 and 9 + (-6) = 3
So, the factors are -6 and +9.
x^{2} + 9x – 6x – 54 = 0
By grouping,
(x^{2} + 9x) + (-6x – 54) = 0
By taking the common factor, we get
x(x + 9) -6(x + 9) = 0
(x – 6)(x + 9) = 0
x – 6 = 0 and x + 9 = 0
x = 6 and x = -9
Example 2 :
x^{2} + x – 72 = 0
Solution :
Product of factors = -72
Sum of factors = 1
Finding the factors of product and sum.
(12) ·(-6) = -72 and 12 + (-6) ≠ 6
(9) · (-8 = -72 and 9 + (-8) = 1
So, the factors are +9 and -8.
x^{2} + 9x – 8x – 72 = 0
By grouping,
(x^{2} + 9x) + (-8x – 72) = 0
By taking the common factor, we get
x(x + 9) -8(x + 8) = 0
(x – 8)(x + 9) = 0
x – 8 = 0 and x + 9 = 0
x = 8 and x = -9
Example 3 :
x^{2} - 4x – 21 = 0
Solution :
Product of factors = -21
Sum of factors = -4
Finding the factors of product and sum.
1 · (-21) = -21 and 1 + (-21) ≠ -20
(-7) · 3 = -21 and (-7) + 3 = -4
So, the factors are -21 and -4.
x^{2} - 7x + 3x – 21 = 0
By grouping,
(x^{2} - 7x) + (3x – 21) = 0
x(x - 7) + 3(x - 7) = 0
(x + 3 )(x - 7) = 0
x + 3 = 0 and x - 7 = 0
x = -3 and x = 7
Example 4 :
x^{2} - x – 6 = 0
Solution :
Product of factors = -6
Sum of factors = -1
Finding the factors of product and sum.
1 · (-6) = -6 and 1 + (-6) ≠ -5
(-3) · 2 = -6 and (-3) + 2 = -1
So, the factors are -3 and 2.
x^{2} - 3x + 2x – 6 = 0
By grouping,
(x^{2} - 3x) + (2x – 6) = 0
x(x - 3) + 2(x - 3) = 0
(x + 2)(x - 3) = 0
Equating each factors to zero.
x + 2 = 0 and x - 3 = 0
x = -2 and x = 3
Example 5 :
x^{2} - 7x – 60
Solution :
Product of factors = -60
Sum of factors = -7
(-20) · 3 = -60 and -20 + 3 ≠ -17
(-12) · 5 = -60 and -12 + 5 = -7
So, the factors are -12 and 5.
x^{2} - 12x + 5x – 60 = 0
x(x - 12) + 5(x - 12) = 0
(x – 12)(x + 5) = 0
x – 12 = 0 and x + 5 = 0
x = 12 and x = -5
Example 6 :
x^{2} + 7x – 60
Solution :
Product of factors = -60
Sum of factors = 7
(-20) · 3 = -60 and (-20) + 3 ‡ -17
12 · (-5) = -60 and 12 + (-5) = 7
So, the factors are 12 and -5.
x^{2} + 12x - 5x – 60 = 0
x(x + 12) - 5(x + 12) = 0
(x + 12)(x - 5) = 0
x + 12 = 0 and x - 5 = 0
x = -12 and x = 5
Example 7 :
x^{2} + 3x – 18
Solution :
Product of factors = 3
Sum of factors = -18
(-9) · 2 = -18 and (-9) + 2 ‡ -7
6 · (-3) = -18 and 6 + (-3) = 3
So, the factors are 6 and -3.
x^{2} + 6x - 3x – 18 = 0
x(x + 6) - 3(x + 6) = 0
(x + 6)(x - 3) = 0
x + 6 = 0 and x - 3 = 0
x = -6 and x = 3
Example 8 :
x^{2} - 7x – 18
Solution :
Product of factors = -18
Sum of factors = -7
(-6) · 3 = -18 and (-6) + 3 ‡ -3
(-9) · 2 = -18 and (-9) + 2 = -7
So, the factors are -9 and 2.
x^{2} - 9x + 2x – 18 = 0
x(x - 9) + 2(x - 9) = 0
(x + 2)(x - 9) = 0
x + 2 = 0 and x - 9 = 0
x = -2 and x = 9
Example 9 :
x^{2} - 12x + 35
Solution :
Product of factors = 35
Sum of factors = -12
1 · 35 = 35 and 1 + 35 ‡ 36
(-7) · (-5) = 35 and (-7) + (-5) = -12
So, the factors are -7 and -5.
x^{2} - 7x - 5x + 35 = 0
x(x - 7) - 5(x - 7) = 0
(x - 5)(x - 7) = 0
x - 5 = 0 and x - 7 = 0
x = 5 and x = 7
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