Sum Product Factoring Calculator

Expression	Sum Target	Product Target	Split Pair	Factored Form
x² + 5x + 6	5	6	2 and 3	(x + 2)(x + 3)
x² - x - 6	-1	-6	-3 and 2	(x - 3)(x + 2)
2x² + 7x + 3	7	6	6 and 1	(2x + 1)(x + 3)
3x² - 8x - 3	-8	-9	-9 and 1	(3x + 1)(x - 3)
x² + 4x + 5	4	5	None	Not factorable over integers

Formula Used

For a monic quadratic, the standard sum-product pattern is x² - Sx + P = (x - m)(x - n), where m + n = S and mn = P.

For a general quadratic ax² + bx + c, the middle-term split uses two numbers p and q such that p + q = b and pq = ac.

The calculator also checks the discriminant Δ = b² - 4ac. It uses the root relation x = (-b ± √Δ) / 2a to report real or complex roots.

Choose Quadratic coefficients when you already know a, b, and c.
Choose Target sum and product when you want numbers that add to one value and multiply to another.
Enter a custom variable symbol if you want a different letter than x.
Press Calculate to show the result directly below the header and above the form.
Review the factor form, roots, discriminant, split pair, and worked method.
Use the export buttons to download a CSV report or a PDF summary.

It means finding two numbers that add to the middle coefficient and multiply to the constant, or to ac for non-monic quadratics.

Use coefficient mode when the quadratic is already written as ax² + bx + c and you want a factoring attempt, roots, and graph.

Use that mode when a problem gives two conditions directly, such as numbers adding to 9 and multiplying to 20.

That message appears when no integer pair matches the required sum and product. The quadratic may still have irrational or complex roots.

It tells you the nature of the roots. Positive gives two real roots, zero gives one repeated root, and negative gives complex roots.

Yes. Negative inputs are supported, and the sign pattern helps determine whether the factor pair has unlike signs or both negative signs.

The graph shows intercepts, turning behavior, and the vertex. It helps you verify whether the calculated roots match the quadratic visually.

The exports summarize the expression, factor form, factor status, roots, discriminant, vertex details, root relations, and the worked steps.