Roman C. answered • 03/24/13
Masters of Education Graduate with Mathematics Expertise
To factor ax2 + bx + c whenever it factors, you want to split the linear term bx as b1x + b2x such that b1b2 = ac and b1 and b2 are integers.
With your case of x2 - 2x - 24, we have b1+b2 = -2 and b1b2 = 1*(-24) = -24.
By factoring -24 we find several choices: (1,-24), (2,-12), (3,-8), (4,-6), (6,-4), (8,-3), (12,-2), and (24,-1)
notice that in the pair (4,-6), that 4+(-6) = -2
Thus we can take b1 = 4 and b2 = -6 and we get
x2 -2x - 24 = x2 + 4x - 6x - 24 = x(x+4) - 6(x+4) = (x - 6)(x + 4)
So the equation x2 - 2x - 24 = 0 becomes (x - 6)(x + 4) = 0.
Thus by Euclid's Lemma, we have x - 6 = 0 or x + 4 = 0, so that x = -4 or x = 6.
Note: Euclid's Lemma states that if you multiply two or more numbers together and the product is 0 then at least one of the numbers is 0.
Roman C.
Slight Correction:
Euclid's Lemma actually says that if a*b is divisible by a prime number p then either a or b is divisible by p.
In the case of ab = 0, the fact that either a or b is 0 is called the "Zero-Divisor Property". It can be shown easily by applying Euclid's Lemma and Euclid's Theorem.
Euclid's theorem simply says that there are infinitely many prime numbers.
03/25/13