Factoring Calculator
Factoring a quadratic expression means rewriting ax squared plus bx plus c as a product of two simpler binomial terms. This is one of the core skills in algebra because a factored form immediately reveals the roots of the equation, simplifies rational expressions, and makes expanding or further manipulating polynomials much easier. The process starts by computing the discriminant, b squared minus 4ac. When the discriminant is positive the quadratic has two distinct real roots and factors into two distinct binomials. When the discriminant is exactly zero the quadratic is a perfect square and factors into one binomial squared. When the discriminant is negative the quadratic has no real roots and cannot be factored over the real numbers. You enter the three coefficients, a for the squared term, b for the linear term, and c for the constant, and the calculator returns the factored form as two binomials, both roots as decimal values, the discriminant, and a verification check showing that the roots satisfy the original equation. The factored form is written in the standard style (ax plus r1)(ax plus r2) divided by a, which handles non-monic quadratics where the leading coefficient is not one. The tool is useful for algebra students checking their working, teachers generating examples, and anyone who needs to factor a quadratic quickly during problem solving. Results assume real-number coefficients and are rounded to four decimal places where needed.
If the discriminant is negative, the quadratic has no real factors. Roots are rounded to 4 decimal places.
How it works
The calculator first evaluates the discriminant D = b squared minus 4ac. If D is negative, no real factored form exists and the calculator says so. If D is zero or positive, the two roots are found using the quadratic formula: x = (-b plus or minus the square root of D) divided by 2a. With roots r1 and r2, the factored form for a monic quadratic (a=1) is (x minus r1)(x minus r2). For a non-monic quadratic (a not equal to 1) the form is a(x minus r1)(x minus r2). The calculator checks whether the roots are clean integers or simple fractions and displays them as such when possible, otherwise showing them to four decimal places.
Worked example
Using the defaults a=1, b=-5, c=6 gives x squared minus 5x plus 6. The discriminant is (-5) squared minus 4 times 1 times 6 = 25 minus 24 = 1. The roots are x = (5 minus 1) divided by 2 = 2 and x = (5 plus 1) divided by 2 = 3. The factored form is therefore (x minus 2)(x minus 3). You can verify by expanding: x squared minus 3x minus 2x plus 6 = x squared minus 5x plus 6.
Related calculators
- Discriminant Calculator: classify the root type of any quadratic using b squared minus 4ac.
- Polynomial Roots Calculator: extend to cubic equations and find up to three real roots.
- Proportion Calculator: solve a/b = c/d for any missing value.
- Percent Error Calculator: verify how close your manual factoring answer is to the correct value.