Quadratic Factored Form Formula

Quadratic factored form is the factored form of a quadratic function is f(x) = a(x.

The Formula

f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2) where r1r_1, r2r_2 are the zeros

When to use: Each factor (x−r)(x - r) equals zero when x=rx = r. So the factored form literally shows you where the parabola crosses the xx-axis—plug in either root and the whole expression becomes zero.

Quick Example

f(x)=3(x−1)(x−4)f(x) = 3(x - 1)(x - 4) The zeros are x=1x = 1 and x=4x = 4. The parabola crosses the xx-axis at these points.

Notation

a(x−r1)(x−r2)a(x - r_1)(x - r_2) where r1r_1, r2r_2 are the xx-intercepts. If the quadratic has a double root, then r1=r2r_1 = r_2 and it becomes a(x−r)2a(x - r)^2.

What This Formula Means

The factored form of a quadratic function is f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2), where r1r_1 and r2r_2 are the zeros (roots) of the function and aa is the leading coefficient.

Each factor (x−r)(x - r) equals zero when x=rx = r. So the factored form literally shows you where the parabola crosses the xx-axis—plug in either root and the whole expression becomes zero.

Formal View

f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2) where r1,r2r_1, r_2 are roots of ff. By Vieta's formulas: r1+r2=−bar_1 + r_2 = -\frac{b}{a} and r1⋅r2=car_1 \cdot r_2 = \frac{c}{a}. If r1,r2∈C∖Rr_1, r_2 \in \mathbb{C} \setminus \mathbb{R}, then r2=r1‾r_2 = \overline{r_1}.

Worked Examples

Example 1

easy
What are the zeros of f(x)=(x−1)(x+4)f(x) = (x - 1)(x + 4)?

Answer

x=1 and x=−4x = 1 \text{ and } x = -4

First step

1
Set each factor to zero: x−1=0x - 1 = 0 gives x=1x = 1; x+4=0x + 4 = 0 gives x=−4x = -4.

Full solution

  1. 2
    The zeros are x=1x = 1 and x=−4x = -4.
  2. 3
    The graph crosses the xx-axis at these points.
In factored form a(x−r1)(x−r2)a(x - r_1)(x - r_2), the zeros are read directly as r1r_1 and r2r_2. This is the most convenient form for finding xx-intercepts.

Example 2

medium
Write a quadratic in factored form with zeros at x=3x = 3 and x=−2x = -2 and passing through (0,−12)(0, -12).

Example 3

easy
Find the axis of symmetry of f(x)=(x+2)(x−8)f(x)=(x+2)(x-8).

Common Mistakes

  • Reading roots with the wrong sign - the root of (x−r)(x-r) is +r+r, so (x+2)(x+2) gives root −2-2.
  • Ignoring the leading factor aa - aa does not change the roots but is needed to match the full function.
  • Setting the whole product equal to a nonzero number and 'solving' each factor - the zero-product trick only works when the product equals 0.

Why This Formula Matters

It exposes the solutions instantly via the zero-product property, which is why factoring is a primary route to solving quadratics. It also lets you reverse-engineer an equation from given intercepts. Recognizing it by "Is the quadratic written as a product of linear factors, and do I want where it equals zero?" — rather than by familiar numbers — is what lets a student tell it apart from standard form and vertex form and factoring (the process) in a mixed problem set.

Frequently Asked Questions

What is the Quadratic Factored Form formula?

The factored form of a quadratic function is f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2), where r1r_1 and r2r_2 are the zeros (roots) of the function and aa is the leading coefficient.

How do you use the Quadratic Factored Form formula?

Each factor (x−r)(x - r) equals zero when x=rx = r. So the factored form literally shows you where the parabola crosses the xx-axis—plug in either root and the whole expression becomes zero.

What do the symbols mean in the Quadratic Factored Form formula?

a(x−r1)(x−r2)a(x - r_1)(x - r_2) where r1r_1, r2r_2 are the xx-intercepts. If the quadratic has a double root, then r1=r2r_1 = r_2 and it becomes a(x−r)2a(x - r)^2.

Why is the Quadratic Factored Form formula important in Math?

It exposes the solutions instantly via the zero-product property, which is why factoring is a primary route to solving quadratics. It also lets you reverse-engineer an equation from given intercepts. Recognizing it by "Is the quadratic written as a product of linear factors, and do I want where it equals zero?" — rather than by familiar numbers — is what lets a student tell it apart from standard form and vertex form and factoring (the process) in a mixed problem set.

What do students get wrong about Quadratic Factored Form?

The procedure for quadratic factored form is the easy part; the trap is reading roots with the wrong sign. Asking "Is the quadratic written as a product of linear factors, and do I want where it equals zero?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Quadratic Factored Form formula?

Before studying the Quadratic Factored Form formula, you should understand: quadratic functions, factoring.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula →
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