Reverse FOIL means moving from a quadratic expression back to multiplied factors. This calculator takes coefficients a, b, and c from ax² + bx + c, finds the real roots with the quadratic formula, and writes the expression as a(x − r₁)(x − r₂).
It works with numeric coefficients and real roots, including repeated and noninteger roots. When the discriminant is negative, the quadratic cannot be written as real linear factors; complex factors would require a different display.
The expression shown uses decimal root values rounded to six places when needed. Exact radical form may be more appropriate for symbolic work involving irrational roots.
For ax² + bx + c, the discriminant is Δ = b² − 4ac.
r₁ = (−b + √Δ) ÷ (2a) and r₂ = (−b − √Δ) ÷ (2a).
When Δ ≥ 0, ax² + bx + c = a(x − r₁)(x − r₂). If Δ = 0, both roots are equal and the factor is repeated.
| Symbol or result | Meaning |
|---|---|
| a | The nonzero leading coefficient multiplying x². |
| b | The signed coefficient multiplying x. |
| c | The signed constant term. |
| Δ | The discriminant, which determines the number of real roots. |
| r₁ and r₂ | The real zeros used in the linear factors. |
| Factored form | The leading coefficient followed by the two root factors. |
For x² + 5x + 6, enter a = 1, b = 5, and c = 6. The discriminant is 5² − 4(1)(6) = 1. The roots are (−5 + 1)/2 = −2 and (−5 − 1)/2 = −3.
The factored result is 1(x + 2)(x + 3). Multiplying gives x² + 3x + 2x + 6 = x² + 5x + 6, confirming all three original coefficients.
Two different real roots produce two different linear factors. A zero discriminant produces one repeated root, such as (x − 2)². A negative discriminant means there is no factorization into real linear factors, though the expression can be factored over complex numbers.
Decimal factors are approximations when roots are irrational. Multiplying rounded factors may produce coefficients close to, rather than exactly equal to, the originals. For formal algebra, preserve exact fractions or radicals. The leading coefficient must remain outside or be distributed correctly; omitting it changes the expression whenever a is not 1.
The roots also connect factorization with the graph. They are the x-coordinates where the parabola meets the horizontal axis. Two roots mean two crossings, one repeated root means the graph just touches, and no real roots means it does not meet that axis. This gives a visual check on the discriminant result.
When coefficients are integers but roots are fractions, the displayed form may use decimal root factors. An equivalent integer-factor form can often be obtained by moving denominators into coefficients. For example, 2x² − 5x + 2 has roots 2 and 0.5, so 2(x − 2)(x − 0.5) is equivalent to (2x − 1)(x − 2). Both expand to the same quadratic.
Reverse FOIL is most useful as a check rather than a substitute for algebraic reasoning. Look first for a common factor in all terms. Factoring out that common value can leave smaller coefficients and make an exact form easier to recognize. Write each intermediate sign clearly when checking.
It refers to multiplying First, Outer, Inner, and Last terms of two binomials. Reverse FOIL seeks factors that expand to the original quadratic.
With a equal to zero there is no x-squared term, so the expression is not quadratic and the quadratic formula does not apply.
It means the quadratic has no real roots and therefore no pair of real linear factors. Complex-number factors still exist.
Both roots are equal, giving a repeated linear factor. The graph touches the x-axis at that root.
That is the leading coefficient a. It is required so expansion reproduces the original x-squared coefficient.
Only when the shown decimals fully represent the roots. Irrational roots are rounded, so use exact radical notation when exact symbolic work is required.
Multiply the factors and leading coefficient. The expanded coefficients of x-squared, x, and the constant should match a, b, and c.