Factoring quadratics is a foundational skill in algebra that helps you simplify expressions and solve equations efficiently. On Khan Academy, learners practice rewriting quadratic expressions as products of binomials to build number sense and prepare for advanced topics. This article explains key ideas with a structured summary, specific practice areas, and common questions to support independent study.
Below is a quick reference table that organizes the main components of factoring simple quadratics and their purpose in problem solving.
| Form of Quadratic | Factoring Strategy | Key Step | Example |
|---|---|---|---|
x^2 + bx + c |
Find two numbers with product c and sum b |
List factor pairs of c |
x^2 + 5x + 6 = (x + 2)(x + 3) |
ax^2 + bx + c, a ≠ 1 |
Multiply a and c, find factor pair that sums to b |
Split middle term and factor by grouping | 2x^2 + 7x + 3 using decomposition |
x^2 − a^2 |
Difference of squares | Apply formula (x − a)(x + a) |
x^2 − 9 = (x − 3)(x + 3) |
| Perfect square trinomial | Recognize square of a binomial | Check first and last terms and middle term | x^2 + 6x + 9 = (x + 3)^2 |
Identify the coefficients in a quadratic expression
Before factoring, clearly label the coefficients a, b, and c in standard form ax^2 + bx + c. On Khan Academy, exercises often highlight these values to guide your first step. Correct identification prevents sign errors later in the process.
Factoring quadratics with a leading coefficient of 1
Find factor pairs that match the middle term
When a = 1, focus on the constant term c and the linear coefficient b. List integer pairs that multiply to c and add to b. Khan Academy provides immediate feedback, helping you verify whether the chosen pair correctly rewrites the quadratic as (x + m)(x + n).
Factoring quadratics with a leading coefficient greater than 1
Use decomposition and grouping carefully
For cases where a ≠ 1, multiply a and c, then find a factor pair of this product that sums to b. Rewrite the middle term using this pair and factor by grouping. This method keeps expressions manageable and aligns with the structured problems on Khan Academy.
Recognizing special patterns in quadratics
Difference of squares and perfect square trinomials
Watch for expressions that fit special patterns, such as x^2 − a^2 or x^2 ± 2ax + a^2. These forms factor quickly without trial and error, saving time during practice. Khan Academy includes pattern recognition drills to reinforce these shortcuts.
Build consistent factoring habits
- Always write the quadratic in standard form before factoring.
- Check whether a greatest common factor exists and factor it out first.
- List factor pairs systematically to avoid missing options.
- Verify your factorization by expanding to ensure you recover the original expression.
- Practice both simple and non-monic quadratics to develop flexibility.
FAQ
Reader questions
How do I know which factor pair to choose when c is positive?
If c is positive, choose a factor pair with the same sign as b; if b is positive, both factors are positive, and if b is negative, both factors are negative.
What do I do when no integer pair adds to b for a non-monic quadratic?
When no integer pair works, the quadratic might be prime over the integers, or you may need to use the quadratic formula later to find exact roots.
Can I factor quadratics with fractions using the same steps?
Yes, you can clear fractions by multiplying by the least common denominator first, then apply standard factoring techniques to the resulting integer coefficients.
Why does Khan Academy show me hints before I try factoring myself?
Hints are designed to nudge you toward the correct strategy, such as identifying the product and sum or recognizing a special pattern, so you learn the method rather than just getting the answer.