EXAMPLES OF SOLVING QUADRATIC EQUATION BY FACTORING

Example 1 :

x² + 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.

x² + 9x – 6x – 54 = 0

By grouping,

(x² + 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 :

x² + 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.

x² + 9x – 8x – 72 = 0

By grouping,

(x² + 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 :

x² - 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.

x² - 7x + 3x – 21 = 0

By grouping,

(x² - 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 :

x² - 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.

x² - 3x + 2x – 6 = 0

By grouping,

(x² - 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 :

x² - 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.

x² - 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 :

x² + 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.

x² + 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 :

x² + 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.

x² + 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 :

x² - 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.

x² - 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 :

x² - 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.

x² - 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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