How this calculator works
A generic rectangle is an area model for algebraic terms, including signed terms that are not literal geometric areas. The diagonal cells contain ax^2 and c. The remaining coefficients solve t^2 - bt + ac = 0. Their sum is b, allowing the middle term to be split for grouping.
Formula
D = b^2 - 4ac; t1 = (b - sqrt(D))/2, t2 = (b + sqrt(D))/2; t1 + t2 = b and t1*t2 = ac.
Worked example
For 3x^2 + 10x + 3, the middle-term coefficients are 1 and 9. Arrange 3x^2 and x in the top row, then 9x and 3 in the bottom row. Grouping gives (3x + 1)(x + 3).
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Frequently asked questions
Must the factors be integers?
No. Real decimal coefficients are accepted. Non-integer roots and factors are displayed approximately.
What if the discriminant is negative?
There is no factorization into real linear factors. The result states this instead of inventing a real box split.
Can the two factors be equal?
Yes. A zero discriminant gives a repeated root and two equal linear factors, apart from the leading multiplier.