Step-by-step explanation:
the point is for such a consideration :
the constant term (like -18 in our example here) is a×b.
and the factor of the x-term (like -7 of -7x) is a+b.
if there are 2 numbers a, b that satisfy these 2 conditions for a trinomial, than it is not prime as it can be factored.
x² - 7x - 18 is not prime.
what factors can create 18 ?
1×18
2×9
3×6
4×4.5
5×3.6
...
-7 = -9 + 2
so, it can be factored into
(x - 9)(x + 2)
x² - 9x - 18 is not prime
a×b = -18
a+b = -9
a = -9 - b
(-9 - b)b = -18
-9b - b² = -18
b² + 9b = 18
b² + 9b - 18 = 0
x² + 9b - 18 = 0
general solution to a quadratic equation :
x = (-b ± sqrt(b² - 4ac))/(2a) = (-9 ± sqrt(9² - 4×1×-18))/(2×1) =
= (-9 ± sqrt(81 + 72))/2 = (-9 ± sqrt(153))/2
x1 = b1 = (-9 + sqrt(153))/2 = 1.684658438... = a2
x2 = b2 = (-9 - sqrt(153))/2 = -10.68465844... = a1
so, we can factor this into
(x + 1.684658438...)(x - 10.68465844...)
or
(x + (-9 + sqrt(153))(x + (-9 - sqrt(153))
these are not rational but still real numbers.
x² + 3x - 18 is not prime
based on the previous 2 examples we can factor this
(x + 6)(x - 3)
x² + 17x - 18 is not prime
based on the same principles this can be factored
(x + 18)(x - 1)
