Enter the three coefficients of a quadratic expression — a, b, and c from ax² + bx + c — and the Generic Rectangle Calculator will factor it using the area model method. You get the factored form, the discriminant, and the roots of the equation, with a visual breakdown of how the rectangle tiles map to each term.
Factored Form
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Discriminant (b² − 4ac)
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Root x₁
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Root x₂
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Product a × c
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Factor Pair (p × q = ac, p + q = b)
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A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to a quadratic equation are called roots or zeros, and there can be 0, 1, or 2 real roots depending on the discriminant.
A generic rectangle is a visual area model used to factor quadratic expressions. The rectangle is divided into four tiles whose areas correspond to the four terms produced when two binomials are multiplied together. By finding which tiles reconstruct the original quadratic, you can identify the binomial factors.
First, multiply a × c to get the product. Then find two integers p and q such that p × q = ac and p + q = b. Place ax², px, qx, and c in the four tiles of the rectangle. Factor out the GCF from each row and column to read off the two binomial factors. This calculator does all those steps automatically.
If a, b, and c share a common factor, you should factor it out before applying the generic rectangle method. For example, 2x² + 4x + 2 simplifies to 2(x² + 2x + 1) first. The calculator works on the coefficients you enter, so divide through by the GCF before inputting the values for the cleanest result.
Not all quadratics factor neatly over the integers. If no integer pair p and q satisfies p × q = ac and p + q = b, the quadratic is said to be irreducible over the integers. The calculator will still return the real roots (if they exist) via the quadratic formula and show the discriminant so you know why integer factoring fails.
For x² + 3x + 2 (a=1, b=3, c=2), you need two numbers that multiply to 1×2=2 and add to 3. Those are 1 and 2. The generic rectangle gives tiles x², 1x, 2x, and 2, factoring to (x + 1)(x + 2). The roots are x = −1 and x = −2.
The generic rectangle method works for any quadratic that factors over the integers. If the discriminant (b² − 4ac) is a perfect square, integer factoring is possible. If not, the quadratic has irrational or complex roots and cannot be expressed as a product of two binomials with integer coefficients.
The discriminant D = b² − 4ac determines the nature of the roots. If D > 0 there are two distinct real roots; if D = 0 there is exactly one repeated real root; if D < 0 the roots are complex (no real solutions). A positive perfect-square discriminant also confirms the quadratic factors nicely over the integers.