Factorisation Calculator

How the Factorisation Calculator Works

Enter your expression using standard algebraic notation and the calculator identifies the factorisation type, applies it, and displays every step. It handles common factors, quadratics, differences of two squares, and grouping. For other advanced series expansions, check our Fourier Series Calculator or standard deviation metrics using the Standard Deviation Calculator.

The calculator verifies your factorisation by expanding the brackets back out. That final expansion check is a habit worth stealing: factorising and expanding are inverse operations, so multiplying your brackets back together and landing on the original expression proves your answer beyond doubt.

Four Techniques and When to Use Them

Almost every factorising question at GCSE and A Level yields to one of four tools, tried in this order:

• Common factor: Pull out anything shared by every single term. Example: 6x² + 9x = 3x(2x + 3)
• Difference of two squares: Anything of the form a² - b² becomes (a + b)(a - b). Example: x² - 25 = (x + 5)(x - 5)
• Quadratic trinomials: x² + bx + c factorises into two brackets using numbers that multiply to c and add to b.
• Grouping: Four-term expressions split into pairs that share a common factor.

The order matters because a hidden common factor disguises the others. For example, 2x² - 50 looks awkward until the 2 comes out, leaving 2(x² - 25), which the difference of two squares finishes as 2(x + 5)(x - 5). Always sweep for the common factor first.

Factorising Quadratics Step by Step

For x² + 7x + 12, find two numbers multiplying to 12 and adding to 7: that is 3 and 4, giving (x + 3)(x + 4). Signs carry critical information:

• A positive constant with a negative middle term (e.g., x² - 7x + 12) means both numbers are negative: (x - 3)(x - 4).
• A negative constant (e.g., x² + x - 12) means one positive and one negative: (x + 4)(x - 3).

When the x² coefficient is bigger than 1, split the middle term. For 2x² + 7x + 3, find numbers multiplying to 2 × 3 = 6 and adding to 7: that is 6 and 1. Rewrite the expression as 2x² + 6x + x + 3, group into 2x(x + 3) + 1(x + 3), and finish with (x + 3)(2x + 1). This splitting method is exactly what the calculator displays for harder quadratics.

Why Factorising Matters Beyond the Classroom

The whole point of factorised form is what it reveals. Set each bracket of a factorised quadratic to zero and the solutions fall out: (x + 3)(2x + 1) = 0 gives x = -3 and x = -0.5, no quadratic formula required. On a graph, those same values are where the curve crosses the x-axis, so factorising sketches the parabola's roots directly.

Algebraic fractions simplify only in factorised form, since common brackets cancel where common individual terms cannot. At A Level, these same skills scale up to factorising cubics using the factor theorem. The revision habit that works: factorise on paper first, then run the same expression through the calculator and compare line by line because seeing exactly where your working diverged fixes misunderstanding faster than any answer key