how would you figure out this equation if you had to factor it out? d^2-20d+99

Question

I am having trouble figuring out this problem d^2-20d+99. I have to factor it out. Please help

Answer 1

For integer factorization, I prefer the product-sum method. This is for factoring a quadratic expression which is in the form ax²+bx+c

 

The method says to find factor pairs of a×c which sum to b

 

in this case, a is 1 and c is 99. the product of 1 and 99 is 1×99=99

 

What are the factor pairs of 99?

1    99

-1  -99

3    33

-3  -33

9     11

-9   -11

 

Do you see a pair of factors that add up to b?

 

(-9)+(-11)=-20

 

So instead of writing b as it is, we're going to split it into this pair of terms:

 

d²-9d-11d+99

 

please notice that I didn't change the value of our expression, I just wrote it in a different way which will prove to be very convenient (and will be every time integer factors exist).

We are going to factor by grouping. I'm going to find the greatest common factor of the first two terms, and of the last two terms.

d²-9d-11d+99

d(d-9)-11(d-9)

 

Now I have two terms, and both terms share the same binomial. I'm going to simply factor out the binomial.

 

d(d-9)-11(d-9)

(d-9)(d-11)

 

 

Answer 2

If the third term were 100, then it would factor into:

(x - 10)(x - 10)

 

so, we can guess that the factors will be just over an under -10

 

( x - 9 )( x - 11 )    !!

 

(replace my "x"s with "d"s

 

Answer 3

If you had to solve the equation by factoring you would set each factor to zero.

 

(d-11)=0 , (d-9)=0

 

d=11,9