Factoring Trinomials Calculator
Enter the coefficients a, b, and c for the trinomial ax2 + bx + c, and the calculator factors it completely over the integers (or shows a decimal factored form for irrational roots). You get the factored expression, the discriminant, both roots, a full step-by-step AC method breakdown, and the factoring method used. The default trinomial 2x2 + 7x + 3 factors to (2x + 1)(x + 3) so you can see the layout immediately.
What is a trinomial and why factor it?
A trinomial is a polynomial with exactly three terms. The standard quadratic trinomial has the form ax2 + bx + c, where a, b, and c are real numbers and a is not zero. Factoring a trinomial means rewriting it as a product of two simpler expressions (usually two linear binomials) instead of a sum of three terms. This is one of the most fundamental skills in algebra because it lets you solve quadratic equations by setting each factor to zero, simplify rational expressions, and understand the graph of a parabola. The x-intercepts of the parabola y = ax2 + bx + c are exactly the values of x that make the factored form equal to zero.
The AC method: factor by grouping
When a is not equal to 1, the standard technique is the AC method. Compute the product a times c. Then search for two integers p and q such that p times q equals ac and p plus q equals b. Once found, rewrite the middle term bx as px + qx, giving you ax2 + px + qx + c. Group the first two and last two terms, factor a common factor out of each group, and the same binomial factor will appear in both groups. Finally, factor that binomial out to get the fully factored form. For example, to factor 2x2 + 7x + 3: ac = 6, and we need two numbers that multiply to 6 and add to 7, which are 1 and 6. Rewrite as 2x2 + 1x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1).
Special cases: perfect squares, difference of squares, and common factors
Three special patterns speed up factoring. A perfect square trinomial arises from squaring a binomial: (px + q)2 = p2x2 + 2pqx + q2. You recognize it when the discriminant equals zero, meaning the first and last terms are perfect squares and the middle term is exactly twice the product of their square roots. A difference of squares appears when b = 0 and c is negative: ax2 - k = (sqrt(a)x + sqrt(k))(sqrt(a)x - sqrt(k)). If the three coefficients share a common integer factor, pull that out first to simplify the remaining trinomial. Always check for a GCF before applying any other method. When c = 0, every term contains an x, so factor x out directly: ax2 + bx = x(ax + b).
When a trinomial does not factor over the integers
Not every trinomial factors neatly into two binomials with integer coefficients. If the discriminant b2 - 4ac is not a perfect square, the roots are irrational and the AC method will not produce an integer pair. In that case, use the quadratic formula to find the two roots x1 and x2, then write the factored form as a(x - x1)(x - x2) with decimal approximations. If the discriminant is negative, the trinomial has no real roots at all and cannot be factored over the real numbers. It can, however, be factored over the complex numbers using imaginary roots, which is a topic in advanced algebra.
Factoring cases at a glance
| Condition | Special case | Factoring method | Result form |
|---|---|---|---|
| b2 - 4ac > 0, integer roots | Two distinct integer roots | AC method / simple factoring | Two different binomials |
| b2 - 4ac = 0 | Repeated root | Perfect square pattern | (px + q)^2 |
| b2 - 4ac > 0, irrational roots | Two irrational real roots | Quadratic formula | a(x - x1)(x - x2), decimal |
| b2 - 4ac < 0 | Complex roots | Not factorable over reals | No real factoring |
| a = 1 | Monic trinomial | Find r, s: r x s = c, r + s = b | (x + r)(x + s) |
| c = 0 | Missing constant | Factor out x | x(ax + b) |
The discriminant b2 - 4ac determines which method applies.
Frequently asked questions
How do I factor a trinomial when a = 1?
When the leading coefficient is 1, the trinomial is x2 + bx + c. Find two numbers r and s such that r times s equals c and r plus s equals b. The factored form is then (x + r)(x + s). For example, x2 + 5x + 6 needs r and s where rs = 6 and r + s = 5, which gives r = 2 and s = 3, so the answer is (x + 2)(x + 3). This is faster than the full AC method because a = 1 means ac = c.
What is the discriminant and what does it tell me?
The discriminant is the expression b2 - 4ac inside the quadratic formula square root. If it is positive, the trinomial has two distinct real roots and factors into two different linear binomials. If it equals zero, there is exactly one repeated root and the trinomial is a perfect square. If it is negative, the roots are complex numbers and the trinomial cannot be factored using real coefficients.
How do I verify my factored answer?
Expand the factored form using the FOIL method (First, Outer, Inner, Last) and check that the result matches the original trinomial. For example, (2x + 1)(x + 3) = 2x2 + 6x + x + 3 = 2x2 + 7x + 3. If the expansion matches, the factoring is correct. This step takes less than a minute and catches sign errors that are easy to make.
Can all trinomials be factored?
No. A trinomial ax2 + bx + c can be factored into two real linear binomials only when its discriminant b2 - 4ac is greater than or equal to zero. If the discriminant is negative, the roots are imaginary and the trinomial is said to be prime over the real numbers. It is still possible to factor such trinomials using complex numbers, but that is not covered by the standard integer-factoring methods.
What is the difference between the AC method and the quadratic formula approach?
The AC method works by finding integer factor pairs and produces a factored form with exact integer coefficients. It only works when the roots are rational numbers. The quadratic formula always produces the roots, but those roots may be irrational decimals or even complex numbers. For homework problems with tidy answers, the AC method is preferred. When the discriminant is not a perfect square, the quadratic formula is the only practical route to a factored form, though the coefficients will be approximate decimals.
What does it mean to factor by grouping?
Factoring by grouping is the second half of the AC method. After you rewrite bx as px + qx (where p + q = b and p x q = ac), you group the four terms into two pairs: (ax2 + px) + (qx + c). Take the greatest common factor out of each pair separately. Both groups will then share a common binomial factor, which you factor out to finish. The result is a product of two binomials.