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Factoring Trinomials

Hi all,

I'm wondering how I can figure out the possible values of b for factorable trinomials in the form ax² + bx + c such as:

2x² + bx + 4

Thanks, Alex Ng 02:08, 8 October 2006 (UTC)

I assume that you want b to be an integer, and the factoring to succeed in the rationals. From the factorization in C, we see that it is required that the discriminant b² – 4ac be a perfect square, say d². Now b² – 4ac = d² iff b² – d² = 4ac. We see that b and d must have the same parity, so p = (b + d) / 2 and q = (b – d) / 2 are integers. Now b = p + q and d = p – q, so we have b² – d² = (p + q)² – (p – q)² = 4pq. The requirement for the discriminant being a perfect square is therefore pq = ac. Each factorization of ac as p × q gives a value p + q for b that results in a factorable quadratic form. For the example, ac = 8, so we have 1 × 8, –1 × –8, 2 × 4, and –2 × –4. This means b can be one of 9, –9, 6, and –6. Since ac always has the factorization a × c, we see that b = a + c is always possible.  --Lambiam^Talk 03:01, 8 October 2006 (UTC)

A simpler way of achieving the same: Write a in the form a₁a₂ and c in the form c₁c₂, any which way you like. Put b = a₁c₂ + a₂c₁. Then ax² + bx + c = (a₁x + c₁)(a₂x + c₂).  --Lambiam^Talk 05:58, 8 October 2006 (UTC)

Oh! I get it now! Thanks Lambiam for the very complete explanation (and the extra add-on for less-mathematically inclined people like me!)

So in summary: to find b in 2x² + bx + 4: a x c = r.

Factor r. (2 x 4 = 8; 8 = 1 x 8, -1 x -8 ... etc), then add the factors (1 + 8 = 9, -1 + (-8) = -9, etc)

Thanks again! Alex Ng 19:59, 8 October 2006 (UTC)

If all else fails, just use the quadratic formula to find roots. --Ķĩřβȳ♥ŤįɱéØ 09:11, 8 October 2006 (UTC)

That quadratic formula seems very confusing, I'll try it out though, thanks Kirbytime! Alex Ng 19:59, 8 October 2006 (UTC)

b is bigger than or equal to 6 * 2^1/2