Start with standard form

A quadratic must be written ax² + bx + c, with a non-zero. Move every term to one side before entering coefficients. For 2x² + 3 = 7x, the standard form is 2x² − 7x + 3, so the inputs are 2, -7 and 3. A sign copied from the other side without changing it produces a different polynomial.

The interface accepts integers because it tests exact rational factorability. Decimal coefficients can be converted to integers by multiplying every coefficient by the same power of ten, provided the decimals are exact. Multiplying the entire polynomial by a non-zero constant does not change its roots.

Greatest common factor first

Before looking for two binomials, factor out the greatest common factor of a, b and c. The calculator reports its positive magnitude. For 6x² − 15x + 9, the common factor is 3, leaving 2x² − 5x + 3. Factoring the simpler primitive quadratic reduces candidate numbers.

A negative leading coefficient can also be handled by taking out a negative common factor so the remaining leading term is positive. The displayed root form keeps the original a rather than rearranging signs for style. In written work, present the form your course expects and verify by expansion.

Discriminant test

The discriminant D = b² − 4ac decides the root type. A negative value gives complex conjugate roots and no factorisation into real linear factors. A zero value gives one repeated rational root. A positive perfect square gives two rational roots. A positive non-square gives irrational real roots.

This creates a fast exact test for integer-coefficient quadratics. Trial factor pairs can be time-consuming, while the discriminant immediately shows whether rational binomials are possible. Do not interpret no rational factorisation as no real roots; a positive non-square discriminant has two real but irrational roots.

Rational roots and factor form

The roots are (-b ± √D)/(2a). When √D is an integer, each root is reduced to a fraction using the greatest common divisor. The polynomial can then be written a(x − r₁)(x − r₂). This form is valid even when roots are fractions.

For 2x² − 7x + 3, D is 25 and the roots are 3 and 1/2. The displayed form 2(x − 3)(x − 1/2) can be rewritten as (x − 3)(2x − 1), which is the usual integer-binomial factorisation. Multiplying the 2 into the fractional-root factor removes the fraction.

Repeated roots

When D equals zero, both root formulas give -b/(2a). The factor form is a(x − r)². Graphically, the parabola touches the x-axis at the repeated root rather than crossing it, provided coefficients are real. The repeated factor has multiplicity two.

A zero discriminant should be exact for integer coefficients. Large coefficients can exceed reliable floating-point integer arithmetic, so keep inputs within ordinary safe ranges and verify by expansion. Symbolic or arbitrary-precision software is preferable for extremely large integers.

The ac grouping method

Another classroom method seeks two integers whose product is ac and whose sum is b. Split bx into those two terms, group the four terms and factor each group. For 2x² − 7x + 3, ac = 6 and the pair -6 and -1 sums to -7.

Writing 2x² − 6x − x + 3 gives 2x(x − 3) − 1(x − 3), so the result is (2x − 1)(x − 3). This method displays integer factors directly and is valuable even when the root form has fractions. The discriminant test tells you whether the search should succeed.

Expand to check

Multiply the factors back out. In a(x − r₁)(x − r₂), expansion gives ax² − a(r₁ + r₂)x + ar₁r₂. Therefore the root sum should equal -b/a and the root product should equal c/a. These are Vieta relationships and provide two quick checks.

A factorisation with the correct constant but wrong middle coefficient is common after a sign error. Check every coefficient, not only roots. Substituting each root into the original polynomial is another valid test, though fractional arithmetic should remain exact.

Difference of squares and special patterns

Some quadratics factor immediately. x² − 25 is (x − 5)(x + 5), using u² − v² = (u − v)(u + v). A perfect-square trinomial such as x² + 6x + 9 is (x + 3)². Recognising these patterns is faster than a general method.

The calculator still reaches the same roots through the discriminant. Use pattern recognition to communicate structure and the calculator to check. A sum of squares such as x² + 25 does not factor over the real numbers, though it factors over complex numbers.

Factors depend on the number system

A polynomial can be irreducible over rational numbers but factor over real or complex numbers. x² − 2 has irrational roots ±√2, so it has no rational linear factors but can be written (x − √2)(x + √2) over the reals. x² + 1 factors only with complex roots.

State the expected number system in an answer. School questions saying factorise often imply integer or rational coefficients. This calculator reports that exact scope instead of replacing irrational roots with rounded decimals and pretending the resulting factors are exact.

Using factors to solve equations

If a factored polynomial equals zero, the zero-product property says at least one factor is zero. Set each linear factor equal to zero to recover the roots. If the expression is not equal to zero, factoring may simplify it but does not by itself create solutions.

In applications, roots still need domain checks. A negative length or time may be mathematically valid but irrelevant. Preserve the original equation and units alongside the algebra so the accepted solution answers the stated problem rather than only the polynomial.

Expand to verify the factors

Multiply the leading coefficient and both displayed linear factors, then collect the squared, linear and constant terms. They must reproduce a, b and c, subject only to shown rounding. If decimal roots produce small discrepancies, return to the exact quadratic formula or radical form instead of treating a rounded factorisation as symbolic equality.