To factor the given quadratic trinomial of the form ak^2+bk+c, rewrite the middle term, bk, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b.
Prime
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We want to completely factor the given expression. Here we have a quadratic trinomial of the form ak^2+bk+c, where |a| ≠1 and there are no common factors. To factor this expression, we will rewrite the middle term, bk, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b.
8k^2-19k+9 ⇕ 8k^2+( -19)k+ 9
We know that a= 8, b= -19, and c= 9. There are now three steps we need to follow in order to rewrite the above expression.
Find a c. Since we have that a= 8 and c= 9, the value of a c is 8* 9=72.
Find factors of a c. Since a c=72, which is positive, we need factors of a c to have the same sign — both positive or both negative — in order for the product to be positive. Since b= -19, which is negative, those factors will need to be negative so that their sum is negative.
c|c|c|c
1^(st)Factor
&2^(nd)Factor
&Sum
&Result
-1
&-72
&-1 + (-72)
&-73
-2
&-36
&-2 + (-36)
&-38
-3
&-24
&-3 + (-24)
&-27
-4
&-18
&-4 + (-18)
&-22
-6
&-12
&-6 + (-12)
&-18
-8
&-9
&-8 + (-9)
&-17
We have not found any factors of 72 that sum up to -19. There are not any integer factors matching our demands. Therefore, the given expression is prime and we cannot factor it using integers.
