For a quadratic there are two special cases that factor very quickly and will save time when recognized.
First case: Difference of Squares
A difference of squares occurs when a quadratic has the form
a2x2 – b2 (1)
where a and b are a positive number. This can then be factored to
(ax – b)(ax + b). (2)
We can see this is true by expanding (2)
= (ax – b)(ax + b)
= (a2x2 + abx – abx – b2)
= (a2x2 – b2)
And it is shown that a2x2 – b2 can be factored into (ax – b)(ax + b).
Examples:
A) x2 – 4 = (x – 2)(x + 2)
We can see that a = 1 and b = 2 in this case.
B) 81x2 – 16 = (9x – 4)(9x + 4) = (3x – 2)(3x + 2)(9x + 4)
When factoring the first time, another difference of squares is formed and so (9x – 4) is factored again.
Second Case: Perfect Squares
A perfect square occurs when b = 2ac in a quadratic, a2 x2 + bx + c2 for some numbers a, b and c. The quadratic can then be factored into
(ax + c)2. (3)
We can see this is true by expanding (3)
= (ax + c)2
= a2x2 + acx + acx + c2
= a2x2 + 2acx + c2
= a2x2 + bx + c2 because b = 2ac
And it is shown that if a quadratic, a2x2 + bx + c2, satisfies b = 2ac then it can be factored into (ax + c)2.
Examples:
A) x2 + 6x + 9 = (x + 3)2
We see that a = 1, c = 3 and b = 6 and b = 2ac
B) 4x2 + 12x + 9 = (2x + 3)2
This article was written for you by Jeremie, one of the tutors with Test Prep Academy.</span
