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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