Explain how to factor the following trinomials forms: x² + bx + c

Question

Explain how
to factor the following trinomials forms: x² + bx + c (Example: x² + 4x + 4) and ax² + bx + c (Example: 2x² + 5x + 3). Be specific and show your answer using both
words and mathematical notation.

Answer 1

For x²+bx+c, notice that (x - r₁)(x - r₂) = x² - (r₁+r₂)x + r₁r₂.

Thus we want integers r₁ and r₂ with a sum -b and product c.

For example, x² + 7x + 10.

Notice that the two integers with a sum -7 and a product of 10 are -2 and -5

Thus we get (x+2)(x+5)

For the more general case ax2+bx+c, if it factors then we can write b = b₁ + b₂ where the following proportion holds:

a:b₁ = b₂:c

or equivalently, b₁b₂ = ac

For example, 2x² + 11x + 12.

We want b₁ + b₂ = 11 and  b₁b₂ = 2*12 = 24

We can take b₁ = 3 and b₂ = 8 and get

2x² + 11x + 12 = 2x² + 3x + 8x + 12 = x(2x + 3) + 4(2x + 3) = (x + 4)(2x + 3)

 

Answer 2

There is formula "ax² + bx + c = a(x-x₁)(x-x₂) , x₁ and x₂ are roots of equation ax² + bx + c = 0 "
x_1,2 = [-b ± √(b² - 4ac)] / 2a

2x² + 5x + 3 = 0 , x_1,2 = [-5 ± √(25 - 24)] / 4 ,
x₁ = (-5 + 1)/4 = -1
x₂ = (-5 - 1)/4 = -6/4 = -3/2
2(x-(-1))(x-(-3/2)) = (x+1)(2x+3)

Answer 3

When you have ax² + bx + c, the quickest way to factor it is to first multiply a by c, and then find two factors that will multiply to equal a*c , add up to b. Then you write (ax + _)(ax + _) with the two factors in the blank spots, then divide it all by a.

Eg. 5x² + 8x + 3 

Here, a*c = 5*3 = 15 and b = 8

Factors that multiply to 15 and add up to 8, are 5 and 3, since 5+3 = 8.

Then write it like so, and you can pull a factor of 5 from the first set of parentheses:

(5x+5)(5x+3) = 5(x+1)(5x+3) but remember we multiplied a by c earlier, so we've actually multiplied the original expression by a (and a = 5), so we need to divide it by a, which we can do to get:

(x+1)(5x+3)

 

Another example:

2x² + 5x + 3

a*c = 2*3 = 6 and b = 5

Factors that add up to 5 and multiply to 6, are: 2 and 3.

So write it as:

(2x + 3)(2x+2) and now divide by a (which is 2): 

(2x + 3)(x+1)

 

Another example

5x² - 16x + 3

a*c = 5*3 = 15 and b = -16

the factors are... (-1)*(-15) = 15 and (-1) + (-15) = -16

(5x - 15)(5x - 1) now divide by a (which is 5)

(x - 3)(5x - 1)