Factored form is the shape a polynomial wears when it is written as a product — and it is the most informative of the three standard shapes, because it displays a quadratic’s roots on sight. This guide puts all three forms of one quadratic side by side, converts in both directions on worked examples, and states the factored form formula that ties the roots to the brackets.

One quadratic, three forms

All three are the same curve: (x − 1)(x − 5) expands to x² − 6x + 5, and (x − 3)² − 4 expands to x² − 6x + 9 − 4 — the same polynomial.

Converting between forms is the daily work of algebra, and a factored form example is only ever finished when expansion returns the standard form — the audit that this guide applies to every conversion below.

The factored form formula

For a quadratic with roots r₁ and r₂ and leading coefficient a, the factored form formula is a(x − r₁)(x − r₂). With a = 1 it shortens to (x − r₁)(x − r₂). Read the formula in the direction that surprises students: the brackets contain the roots with their signs flipped, because a bracket (x − r) is zero exactly when x = r. So (x − 1)(x − 5) has roots +1 and +5, and a quadratic whose roots are −6 and 4 factors as (x + 6)(x − 4). The formula also runs the whole conversion backwards: expand a(x − r₁)(x − r₂) and the standard form’s coefficients are forced — b = −a(r₁ + r₂) and c = a × r₁ × r₂ — which is precisely the product-and-sum hunt from trinomial factoring, seen from the other side.

Standard to factored: three conversions

By the pair hunt. x² + 2x − 24: find a pair multiplying to −24 and adding to 2 — 6 and −4 — so x² + 2x − 24 = (x + 6)(x − 4), roots −6 and 4 ✓.

GCF first. 2x² + 10x + 12: the GCF is 2, giving 2(x² + 5x + 6), and the bracket hunts to (x + 2)(x + 3), so the completely factored form is 2(x + 2)(x + 3). Dropping the 2 is the classic conversion error — the factored form must expand back to the whole original, and (x + 2)(x + 3) alone expands to x² + 5x + 6, half the polynomial.

GCF only. 3x² − 12x has no constant term; its factored form is 3x(x − 4) — a product of three factors, two of them monomials. Factored form does not require brackets everywhere; it requires a product, completely split.

Factored to standard: expansion, worked

(x + 7)(x − 3): multiply out — x² − 3x + 7x − 21 = x² + 4x − 21. The middle coefficient, 4, is the sum of the roots-with-flipped-signs (7 and −3), as the formula promised.

4(x − 2)(x + 5): expand the brackets first — (x − 2)(x + 5) = x² + 3x − 10 — then multiply by 4: 4x² + 12x − 40. Doing the 4 last keeps the numbers small; distributing it first is legal but noisier, and noise is where signs go to die.

The conversion table, both directions

Six conversions, each verified in reverse: every factored form here expands to its standard-form partner, and vice versa.

The Algebra Calculator moves between these forms on any coefficients — its quadratic mode reports the roots that factored form displays, which makes hand conversions auditable line by line.

What factored form reveals that standard form hides

• The roots, instantly: read each bracket, flip the sign. No computation at all.
• The y-intercept, almost instantly: set x = 0 in the brackets and multiply — (0 + 6)(0 − 4) = −24 matches the constant of x² + 2x − 24.
• The sign of the curve between roots: for (x + 6)(x − 4), the product is negative exactly when the brackets disagree in sign — between x = −6 and x = 4 — so the parabola dips below the axis there and nowhere else.
• Multiplicity: a repeated bracket like (x − 3)² means a repeated root — the curve touches the axis at x = 3 and turns around instead of crossing.
• The axis of symmetry: the roots are symmetric about the vertex’s x-coordinate, so averaging them locates it — for (x + 6)(x − 4) the roots −6 and 4 average to −1, and the parabola’s axis is the line x = −1. Factored form quietly contains the vertex’s address; standard form makes you compute it with −b ÷ 2a.

The three conversion mistakes

• Root signs read unflipped. (x − 5) means root +5. More marks are lost here than in all the factoring itself.
• The leading coefficient dropped. 2x² + 10x + 12 is 2(x + 2)(x + 3), not (x + 2)(x + 3) — expand to audit.
• Confusing factored form with vertex form. (x − 3)² − 4 is not a product; it shows the vertex, not the roots. The names name what they show.

Factored form beyond quadratics

The idea does not stop at degree two — a polynomial is in factored form whenever it is written as a product of lower-degree factors, split as far as the number system allows. The cubic from grouping practice, x³ − 2x² + 4x − 8, factors as (x − 2)(x² + 4): completely factored over the reals, because x² + 4 (a sum of squares) splits no further without complex numbers. “Completely factored” always carries that quiet qualifier — over the integers, over the reals, or over the complex numbers, the stopping point differs, and honest work says which one it means.

The formula a(x − r₁)(x − r₂) also explains a conversion that puzzles people: 6x² + 11x + 3 has roots −1/3 and −3/2, so its formula form is 6(x + 1/3)(x + 3/2). Multiply the 6 into the brackets — 3 into the first, 2 into the second — and the fractions clear: (3x + 1)(2x + 3), the same factored form the X method produces. Fractional roots inside the formula and integer-looking brackets are not rival answers; the leading coefficient is simply parked in different places, and expansion proves the two identical.

The three forms again, with a leading coefficient

Everything above used a = 1; the forms survive a ≠ 1 unchanged, with the coefficient riding along in each. Take 2x² − 8x + 6. Standard form: 2x² − 8x + 6, y-intercept 6. Factored form: shed the GCF and hunt the bracket — 2(x² − 4x + 3) = 2(x − 1)(x − 3) — roots 1 and 3, read straight off. Vertex form: 2(x − 2)² − 2, vertex (2, −2). All three verified against each other: 2(x − 1)(x − 3) = 2(x² − 4x + 3) = 2x² − 8x + 6 ✓, and 2(x − 2)² − 2 = 2(x² − 4x + 4) − 2 = 2x² − 8x + 6 ✓. Note where the 2 lives in each shape: multiplied through in standard form, parked outside the brackets in factored form, and scaling the squared bracket in vertex form. Losing track of that coefficient mid-conversion is, again, the most common way the three forms stop agreeing — and the expansion audit is how you catch it.

The same parking trick closes the loop with the solving bank: 5x² + 42x + 49 has roots −7/5 and −7, so the formula writes it as 5(x + 7/5)(x + 7) — and pushing the 5 into the first bracket clears the fraction to give (5x + 7)(x + 7), exactly the factored form the solving routine produced. Wherever you meet a factored form, the factored form formula can regenerate it from the roots alone; that is the audit trail that makes the form trustworthy.

Standard form is for computing, vertex form is for locating the turn, factored form is for reading the roots — and converting fluently among the three is most of what “being good at quadratics” actually consists of.

Method: every conversion in this guide was verified symbolically in both directions — factored forms re-expanded to their standard forms, and standard forms re-factored and compared.