EXAMPLES OF FACTORING QUADRATIC EXPRESSIONS

Example 1 :

5x² – 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 :

= 5x² - 5x - 3x + 3

By grouping,

= (5x² – 5x) + (-3x + 3)

Step 4 :

By taking the common factor, we get

= 5x(x - 1) - 3(x – 1)

= (x – 1)(5x - 3)

Example 2 :

11x² – 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 :

= 11x² - 11x + 2x - 2

By grouping,

= (11x² – 11x) + (2x – 2)

Step 4 :

By taking the common factor, we get

= 11x(x - 1) + 2(x – 1)

= (x - 1)(11x + 2)

Example 3 :

3x² – 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 :

= 3x² - 9x + 2x – 6

By grouping,

= (3x² - 9x) + (2x - 6)

Step 4 :

By taking the common factor, we get

= 3x(x - 3) + 2(x – 3)

= (x - 3)(3x + 2)

Example 4 :

2x² – 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 :

= 2x² - 6x + 3x – 9

By grouping,

= (2x² - 6x) + (3x - 9)

Step 4 :

By taking the common factor, we get

= 2x(x - 3) + 3(x - 3)

 = (x - 3)(2x + 3)

Example 5 :

3x² + 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

= 3x² + 12x - 2x – 8

By grouping,

= (3x² + 12x) + (-2x - 8)

By taking the common factor, we get

= 3x(x + 4) - 2(x + 4)

= (x + 4)(3x – 2)

Example 6 :

2x² + 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.

= 2x² + 12x - 3x – 18

By grouping,

= (2x² + 12x) + (-3x – 18)

By taking the common factor, we get

= 2x(x + 6) - 3(x + 6)

 = (2x – 3)(x + 6)

Example 7 :

2x² + 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.

= 2x² + 14x - 3x – 21

By grouping,

= (2x² + 14x) + (-3x - 21)

By taking the common factor, we get

= 2x(x + 7) - 3(x + 7)

 = (2x - 3)(x + 7)

Example 8 :

15x² + 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.

= 15x² - 5x + 6x – 2

By grouping,

= (15x² - 5x) + (6x – 2)

By taking the common factor, we get

= 5x(3x - 1) + 2(3x - 1)

 = (5x + 2)(3x - 1)

Example 9 :

21x² – 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.

= 21x² + x - 63x – 3

By grouping,

= (21x² + x) + (-63x - 3)

By taking the common factor, we get

= x(21x + 1) - 3(21x + 1)

= (x - 3)(21x + 1) 

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