Factoring x^2 + bx + c
A.SSE.2
Read the standard text
A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, to factor 3x(x − 5) + 2(x − 5), students should recognize that the "x − 5" is common to both expressions being added, so it simplifies to (3x + 2)(x − 5); or see x^4 − y^4 as (x^2)^2 − (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 − y^2)(x^2 + y^2).
Section 1 · LearnLearn the idea
In this lesson you will: Factor x^2+bx+c by finding two numbers with product c and sum b; Use the signs of b and c to know what kind of pair to look for; Read the zeros of the expression from its factored form.
Two numbers, one product, one sum
Multiply (x+3)(x+5) and you get x^2+8x+15. Look at where those numbers went: 3*5=15 landed in the constant, and 3+5=8 landed in the middle.
That is the whole method, backwards. To factor x^2+bx+c, find two numbers whose product is c and whose sum is b. Write them into (x+ )(x+ ) and you are done.
The signs tell you what to look for
Before hunting for the pair, read the two signs. They narrow the search to one short list.
If c is positive, the two numbers have the same sign, and that sign is the sign of b. If c is negative, the two numbers have opposite signs, and the one with the larger absolute value carries the sign of b.
What the factored form shows you
x^2-9x+14=(x-2)(x-7). Those two forms are the same expression, but they answer different questions. The first tells you the value at x=0 at a glance; the second tells you where the expression is zero.
A product of real numbers is zero only when one of the factors is zero, so (x-2)(x-7) is zero at x=2 and at x=7 and nowhere else. That is what A.SSE.3a means by factoring to reveal the zeros — and it is why Chapter 16 opens by factoring.
Common mistake — Getting the pair right and the signs wrong
To factor x^2-2x-24, a student finds that 4 and 6 multiply to 24 and differ by 2, then writes (x-4)(x+6).
The numbers were the right pair. Only the placement of the signs went wrong, which is why this error survives a quick glance — the answer looks like a properly factored expression.
But (x-4)(x+6)=x^2+2x-24. The middle term came out +2x instead of -2x. Because b is negative here, the larger number must be the negative one: (x+4)(x-6).
How to catch it: Add the two numbers you wrote, signs included, and compare with b. Here -4+6=+2, and the expression needs -2, so the signs are backwards. This check takes a second and catches every sign slip of this kind.
Section 2 · DoPractice this standard
These published practice sets are scored on the student site and cover A.SSE.2 directly.