What is a Factoring Calculator?
This calculator factors quadratic expressions and shows the steps.
How it's calculated
x² + (a+b)x + ab = (x+a)(x+b)
The key is finding two numbers that add to give b and multiply to give c.
Step-by-step example
Factoring x² + 7x + 12
- Step 1 Find two numbers that multiply to 12 and add to 7
- Step 2 3×4=12, 3+4=7 — the numbers are 3 and 4
- Step 3 Factored form: (x+3)(x+4)
Interpretation: This expression equals zero when x=−3 or x=−4.
Why factor at all?
Turning an expression into a product lets you immediately find where it equals zero. For two numbers multiplied together to equal zero, one of them must be zero — so (x−2)(x+3) = 0 instantly tells you x = 2 or x = −3. Left in expanded form (x² + x − 6 = 0), the answer isn't obvious at all. Converting to a product is what reveals the answer.
Good to know
- This same principle scales up dramatically — prime factorization is the foundation of modern cryptography. Multiplying two large primes is easy, but factoring the resulting huge number back into its primes is extremely hard.
- Factoring a 300-digit number would take current computers thousands of years — the reason online banking and HTTPS security remain safe.
- Shor's algorithm, designed for quantum computers, can factor large numbers efficiently — which is exactly why current cryptography faces a looming threat, and why 'post-quantum cryptography' is actively being developed.
- Memorizing patterns like the difference of squares (a²−b² = (a+b)(a−b)) dramatically speeds up mental math — 98×102 = (100−2)(100+2) = 10,000−4 = 9,996.
- Not every polynomial factors over the rational numbers, but every polynomial factors completely into linear terms over the complex numbers — this is the Fundamental Theorem of Algebra.
Frequently asked questions
Q. What if an expression doesn't factor nicely?
Some expressions don't factor over rational numbers — use the quadratic formula instead.
Q. How do I find the right two numbers quickly?
List factor pairs of the constant term, then check which pair sums to the coefficient of the middle term.
Q. What if the leading coefficient isn't 1?
For ax²+bx+c, multiply a×c, find two numbers with that product summing to b, then use grouping (the 'ac method').