FACTORING TRINOMIALS WHEN A IS NOT 1

The general form any quadratic equation will be in the form

ax2 + bx + c

To factorize a quadratic polynomial, we have to check whether the coefficient of x2 is 1 or not equal to 1. 

Here we see examples on factoring quadratic polynomial when the coefficient x2 is, that is a is not equal to 1.

If the quadratic polynomial is in the form,

ax2 + bx + c ---> Both factors are positive

ax2 - bx + c ---> Both factors are negative

ax2 - bx - c ---> The large factor will be negative

ax2 + bx - c ---> The small factor will be negative

Factoring trinomials when a is not 1.

Problem 1:    

7m² + 6m - 1

Solution :

= 7m² + 6m - 1

Factors of -7

-1 and 7

Sum

6

Product

-7

= 7m² + 7m – m – 1

= 7m (m + 1) – 1 (m + 1)

= (7m - 1) (m + 1)

Problem 2 :    

3k² - 10k + 7

Solution :

Factors of 21

-3 and -7

Sum

-10

Product

21

= 3k² - 10k + 7

= 3k² - 3k - 7k + 7

= 3k(k - 1) - 7(k - 1)

= (3k - 7) (k - 1)

Problem 3 :    

5x² - 36x - 81

Solution :

Factors of -405

-45 and 9

Sum

-36

Product

-405

= 5x² - 36x - 81

= 5x² - 45x + 9x – 81

= 5x(x - 9) + 9(x - 9)

= (5x + 9) (x - 9)

Problem 4 :    

2x² - 9x - 81

Solution :

Factors of -162

-18 and 9

Sum

-9

Product

-162

= 2x² - 9x - 81

= 2x² - 18x + 9x - 81

= 2x(x - 9) + 9(x - 9)

= (2x + 9) (x - 9)

Problem 5 :    

3n² - 16n + 20

Solution :

Factors of 60

-10 and -6

Sum

-16

Product

60

= 3n² - 16n + 20

= 3n² - 6n – 10n + 20

= 3n(n - 2) – 10(n - 2)

= (3n - 10) (n - 2)

Problem 6 :    

2r² + 7r - 30

Solution :

Factors of 60

12 and -5

Sum

-60

Product

7

= 2r² + 7r - 30

= 2r² +12r – 5r – 30

= 2r(r + 6) – 5(r + 6)

= (2r - 5) (r + 6)

Problem 7 :    

5k² + 8k + 80

Solution :

5k² + 8k + 80 is not factorable.

Problem 8 :

5x² - 14x + 8

Solution :

Factors of 60

-10 and -4

Sum

-14

Product

40

= 5x² - 14x + 8

= 5x² - 10x – 4x + 8

= 5x(x - 2) – 4(x - 2)

= (5x - 4) (x - 2)

Problem 9 :

7p² - 20p + 12

Solution :

Factors of 84

-14 and -6

Sum

-20

Product

84

= 7p² - 20p + 12

= 7p² - 14p – 6p + 12

= 7p(p - 2) – 6(p - 2)

= (7p - 6) (p - 2)

Problem 10 :  

3v² + 14v - 49

Solution :

= 3v² + 14v - 49

= 3v² + 21v – 7v – 49

= 3v(v + 7) – 7(v + 7)

= (3v - 7) (v + 7)

Problem 11 :  

7x² - 26x - 45

Solution :

= 7x² - 26x - 45

= 7x² - 35x + 9x – 45

= 7x(x - 5) + 9(x - 5)

= (7x + 9) (x - 5)

Problem 12 :  

5p² - 52p + 20

Solution :

= 5p² - 52p + 20

= 5p² - 50p – 2p + 20

= 5p(p - 10) – 2(p - 10)

= (5p - 2) (p - 10)

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