Explanation:
x² + 72 = -17x
x² + 17x + 72 = 0
factorization means to find a way to express this as
(x + a)(x + b) = 0
because then -a and -b are the points of zero, as when at least one factor is zero, then the whole expression is zero.
let's multiply the general expression and then compare the factors of the individual terms :
x² + bx + ax + ab = 0
x² + (a+b)x + ab = 0
and our specific equation is
x² + 17x + 72 = 0
the factor of x² is 1, of course.
the factor of x is 17 = a+b
the constant is 72 = ab
we see right away that 8×9 = 72, and 8+9 = 17
so, 8 and 9 are our solutions for a and b.
and the factorization looks like
(x + 8)(x + 9) = 0
and the solutions to the equation are therefore
x = -8 and
x = -9
FYI
now, if we don't see the solution right away, we need to formally solve the 2 equations with the 2 variables a, b.
again, we have
17 = a + b
72 = ab
from the first equation we get e.g.
a = 17 - b
and we use that in the second equation
72 = (17 - b)b = 17b - b²
-b² + 17b - 72 = 0
and now we need to solve this quadratic equation.
ultimately, there is the general solution for such an equation :
b = (-m ± sqrt(m² - 4ln))/(2l)
in our case
l = -1
m = 17
n = -72
b = (-17 ± sqrt(17² - 4×-1×-72))/(2×-1) =
= (-17 ± sqrt(289 - 288))/-2 =
= (-17 ± sqrt(1))/-2 = (-17 ± 1)/-2
b1 = (-17 + 1)/-2 = -16/-2 = 8
b2 = (-17 - 1)/-2 = -18/-2 = 9
there we have our solutions that we saw originally directly.
a1 and a2 are based on the equations also 9 and 8 (just contrary to b), so that 8 and 9 are our numbers for the factorization, leading to the solutions in x as above.
we could have done that approach with the original equation in x to, but the problem asked specifically for factorization.