Factoring ax² + bx + c

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

For a monic quadratic, factoring means finding two numbers with a specified sum and product. For a nonmonic quadratic, split the middle term so grouping becomes possible. Not every integer-coefficient quadratic has integer factors.

Method

(x+p)(x+q)=x^2+(p+q)x+pq.

For $ax^2+bx+c$ , seek numbers whose product is $ac$ and sum is $b$ . If none exist among integers, use the discriminant or quadratic formula rather than forcing integer factors.

Worked example

Factor $6x^2+11x+3$ . The product $ac=18$ and sum $b=11$ suggest $9$ and $2$ :

6x^2+9x+2x+3=3x(2x+3)+(2x+3).

Hence $(3x+1)(2x+3)$ . The two zeros are $-1/3$ and $-3/2$ .

Common mistake

Matching only the constant term is insufficient. Check the middle coefficient as well as the leading coefficient.

Check your understanding

Factor $x^2-x-6$ .

Show answer

$(x-3)(x+2)$ : the constants sum to $-1$ and multiply to $-6$ .

Explore

Factors and zeros

Try this. Move the roots together and apart. Match the expanded coefficients to their sum and product, then compare touching with crossing the axis.

First root-2Second root2

(x − (-2))(x − (2)) = x² − (0)x + (-4). Zeros: -2, 2. Simple roots: the graph crosses at each zero.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Which factorization equals

x^2+5x+6

(x+2)(x+3)

(x+1)(x+6)

(x−2)(x−3)

Hint 1 · Find a starting point

Seek two numbers whose product is 6.

Hint 2 · Take the next step

Their sum must also equal the middle coefficient, 5.

Show the reasoning

Answer: (x+2)(x+3)

2 × 3 = 6 and 2 + 3 = 5.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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