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Calculus Readiness · Axiom Academy
Unwinding ax^2 + bx + c back into two binomials — by hunting for the pair of numbers that multiply and add just right. 1. Factoring Is Finding a Rectangle A trinomial like x^2 + 5x + 6 is literally an area : an square, five strips, and six squares. Factoring asks one question — can those pieces be repacked into a single rectangle? If they can, the rectangle's two sides are the factors. The six units split as — that fixes the corner …and those same 2 and 3 are the strips that make 5x So the method is a hunt. List the factor pairs of c , check each pair's sum , and stop at the one that equals b . Watch the search run for x^2 + 9x + 20 : only 4 + 5 lands on the target. The signs tell you where to look before you even start. Since and b = r + s , the signs of c and b pin down the signs of the pair: both negative. (-3)(-5)=15 , -3-5=-8 , so (x-3)(x-5) . opposite signs. (+5)(-3)=-15 , 5-3=2 , so (x+5)(x-3) . If the leading coefficient isn't 1, the pair must now multiply to (not just c ) while still adding to b . Use that pair to split the middle term , drop the four terms into a 2×2 box, and read the factors off the edges. Here's 6x^2 + 11x - 10 , where ac = -60 and the winning pair is 15 and -4 . Split: 11x = 15x - 4x , so 6x^2 + 11x - 10 = 6x^2 + 15x - 4x - 10 — the four cells of the box. Group: pull the GCF from each row, 3x(2x+5) - 2(2x+5) , then factor out the shared (2x+5) to land on (3x-2)(2x+5) . 4x^2+20x+24 = 4(x^2+5x+6) = 4(x+2)(x+3) — smaller numbers, easier search.
This is the written version of the interactive lesson above. See the full Calculus Readiness course.