What are quadratic trinomials?
A quadratic trinomial is a polynomial of degree 2 with exactly three terms, typically written as:
ax² + bx + c
where a, b, c are real number coefficients and a ≠ 0 (the squared term must be present). The term a is called the leading coefficient.
To factor a quadratic trinomial means finding two linear binomials whose product equals the original trinomial:
ax² + bx + c = α(x − r₁)(x − r₂)
where r₁ and r₂ are the roots. Factoring a quadratic trinomial is the reverse of multiplying two binomials (FOIL method).
How to factor quadratics with this factoring trinomials calculator?
- Enter coefficients a, b, c of your trinomial ax² + bx + c. Use the live preview to confirm the polynomial looks correct.
- Click the quick-example buttons to try well-known examples instantly.
- Optionally enable "Show step-by-step solution" to see the full factoring process — either via the AC grouping method (for integer coefficients) or the quadratic formula.
- Click Calculate. The calculator returns:
- The factored form
- The roots (zeros) of the trinomial
- The discriminant value and what it tells you
- A step-by-step breakdown (if enabled)
Tip: You can share or revisit any calculation using the unique link generated after each result.
How to factor trinomials
There are three main methods for factoring a quadratic trinomial ax² + bx + c:
1. Quadratic formula
The most universal method. Calculate the discriminant D = b² − 4ac, then find roots:
x = (−b ± √D) / (2a)
- D > 0: two distinct real roots → trinomial factors as a(x − r₁)(x − r₂)
- D = 0: one repeated root → perfect square trinomial a(x − r)²
- D < 0: complex roots → trinomial cannot be factored over real numbers
2. Recognizing a perfect square trinomial
A trinomial ax² + bx + c is a perfect square when D = 0, i.e., b² = 4ac. It factors as a(x − r)² where r = −b/(2a).
Example: x² − 6x + 9 = (x − 3)² (since D = 36 − 36 = 0)
3. Grouping method (AC method)
Works best for integer coefficients. See the full explanation in the next section.
Factoring trinomials steps
Here is the complete algorithm for factoring ax² + bx + c step by step:
- Identify a, b, c — the leading coefficient, middle coefficient, and constant term.
- Calculate the discriminant: D = b² − 4ac
- Determine the type:
- D > 0 → two real roots, trinomial is factorable
- D = 0 → perfect square
- D < 0 → complex roots, not factorable over ℝ
- Find the roots using x = (−b ± √D) / (2a)
- Write the factored form: a(x − r₁)(x − r₂)
Example: Factor 3x² + 10x + 3
- D = 10² − 4·3·3 = 100 − 36 = 64
- √D = 8
- x₁ = (−10 + 8) / 6 = −1/3 → factor: (3x + 1)
- x₂ = (−10 − 8) / 6 = −3 → factor: (x + 3)
- Result: 3x² + 10x + 3 = (3x + 1)(x + 3)
Factoring trinomials by grouping: tips
The AC method (grouping method) is especially useful when all coefficients are integers:
- Compute ac (the product of the leading coefficient and the constant).
- Find two integers p and q such that: p · q = ac and p + q = b
- Rewrite the middle term: ax² + bx + c = ax² + px + qx + c
- Factor by grouping: factor out from each pair and extract the common binomial.
Example: Factor x² + 8x + 12
- ac = 1 · 12 = 12. Find p, q: 2 · 6 = 12 and 2 + 6 = 8 ✓
- Rewrite: x² + 2x + 6x + 12
- Group: x(x + 2) + 6(x + 2)
- Factor: (x + 2)(x + 6)
Tip: When looking for p and q, systematically list factor pairs of ac (both positive and negative) and check which pair sums to b. Our calculator does this automatically!
FAQs
What if the discriminant is negative?
If D < 0, the trinomial has no real roots and cannot be factored over the real numbers. However, it can be factored over the complex numbers: a(x − z₁)(x − z₂) where z₁, z₂ are complex conjugates. Our calculator shows the complex roots in this case.
Can I factor trinomials with non-integer (decimal or fractional) coefficients?
Yes! Enter any real-number coefficients — the calculator uses the quadratic formula, which works for any values of a, b, c (as long as a ≠ 0). The AC grouping method is only shown for integer coefficients.
What is a perfect square trinomial?
A perfect square trinomial is one that can be written as (px + q)². It occurs exactly when D = 0. Examples: x² + 4x + 4 = (x+2)², 9x² − 12x + 4 = (3x − 2)².
How does this calculator relate to the quadratic formula calculator?
They are closely related — both use the discriminant and the same formula for roots. The factoring calculator additionally displays the factored polynomial form a(x − r₁)(x − r₂), making it ideal for algebra homework and equation solving.
Does this calculator support US and metric measurements?
Factoring trinomials is a purely algebraic operation — coefficients are dimensionless numbers, so no units or conversions apply. If you need to factor an expression that arises from a real-world formula involving US or metric quantities, simply extract the numerical coefficients and enter them here.