Factoring ax2+bx+c, c is positive
Algebra
Objective: To factor general quadratic trinomials with integral coefficients.
In the last lessons, we learned how to factor questions in which the coefficient of the squared term was 1. For example:
x2 – 15x + 56 = (x – 8)(x – 7) and y2 – 5y – 24 = (y + 3)(y – 8)
In the first example above the mental question was, “what multiplies to 56 and adds to 15?” The other example’s questions was,
“what multiplies to 24 and subtracts to 5. However, this only works when the squared term has a coefficient of 1. Now, we must
factor trinomials like 3x2 + 7x + 2. We can longer ask ourselves, “what multiplies to 2 and adds to 7?”, since the squared term has a
coefficient other than 1. There are two ways to factor this trinomial: guess and check, and group factoring.
Example 1:
Guess and Check
Group Factoring
2
2
Factor: 3x + 7x + 2
Factor: 3x + 7x + 2
What mult to 6 and adds to 7? 6 and 1
2
(3x + 2)(x + 1)
3x + 1x + 6x + 2
Split it down the middle
(3x + 1)(x + 2)
x(3x + 1) + 2(3x + 1)
Monomial factor
(3x + 1)(x + 2)
Create two binomials
When the squared term is prime and the constant is prime, the guess and check method is quite efficient. However, when the squared
term is composite and the constant term is composite, the group factoring method will be more efficient.
Example 2:
Guess and Check
Group Factoring
2
Factor: 9p + 18p + 8
2
What mult to 72 and adds to 18? 6 and 12
2
Factor: 9p + 18p + 8
(9p + 1)(p + 8)
9p + 6p + 12p + 8
Split it down the middle
(9p + 8)(p + 1)
3p(3p + 2) + 4(3p + 2)
Monomial factor
(9p + 4)(p + 2)
(3p + 2)(3p + 4)
Create two binomials
(9p + 2)(p + 4)
(3p + 2)(3p + 4)
(3p + 4)(3p + 2)
Example 3:
Guess and Check
Group Factoring
Factor: 14x2 – 17x +5
Factor: 14x2 – 17x + 5
What mult to 70 and adds to 17? 7 and 10
2
(14x – 1)(x – 5)
14x – 7x – 10x + 5
Split it down the middle
(14x – 5)(x – 1)
7x(2x – 1) – 5(2x – 1)
Monomial factor
(7x – 1)(2x – 5)
(2x – 1)(7x – 5)
Create two binomials
(7x – 5)(2x – 1)
Factor completely.
1.
3c2 – 8c + 5
2.
7x2 + 8x + 1
3.
5x2 – 17x + 6
4.
3m2 + 11m + 6
5.
2x2 – 15x + 7
6.
2p2 + 7p + 3
7.
2g2 – 7g + 6
8.
4t2 + 8t + 3
9.
6h2 + 17h + 10
10.
3m2 – 11m + 10
11.
6u2 – 17u + 12
12.
7k2 + 18k + 11
13.
15w2 + 26w + 8
14.
8a2 – 17a + 2
15.
4x2 + 27x + 35