Quadratic Factored Form Formula

The Formula

f(x) = a(x - r_1)(x - r_2) where r_1, r_2 are the zeros

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

Quick Example

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

Notation

a(x - r_1)(x - r_2) where r_1, r_2 are the x-intercepts. If the quadratic has a double root, then r_1 = r_2 and it becomes a(x - r)^2.

What This Formula Means

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

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

Formal View

f(x) = a(x - r_1)(x - r_2) where r_1, r_2 are roots of f. By Vieta's formulas: r_1 + r_2 = -\frac{b}{a} and r_1 \cdot r_2 = \frac{c}{a}. If r_1, r_2 \in \mathbb{C} \setminus \mathbb{R}, then r_2 = \overline{r_1}.

Worked Examples

Example 1

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

Solution

  1. 1
    Set each factor to zero: x - 1 = 0 gives x = 1; x + 4 = 0 gives x = -4.
  2. 2
    The zeros are x = 1 and x = -4.
  3. 3
    The graph crosses the x-axis at these points.

Answer

x = 1 \text{ and } x = -4
In factored form a(x - r_1)(x - r_2), the zeros are read directly as r_1 and r_2. This is the most convenient form for finding x-intercepts.

Example 2

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

Common Mistakes

  • Forgetting to include the leading coefficient a when writing factored form
  • Getting the sign wrong—the roots of (x - 3)(x + 2) are x = 3 and x = -2, not x = -3 and x = 2
  • Assuming all quadratics can be factored over the integers

Why This Formula Matters

Finding zeros is one of the most common tasks in algebra. Factored form also makes it easy to determine the sign of the function on different intervals.

Frequently Asked Questions

What is the Quadratic Factored Form formula?

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

How do you use the Quadratic Factored Form formula?

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

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

a(x - r_1)(x - r_2) where r_1, r_2 are the x-intercepts. If the quadratic has a double root, then r_1 = r_2 and it becomes a(x - r)^2.

Why is the Quadratic Factored Form formula important in Math?

Finding zeros is one of the most common tasks in algebra. Factored form also makes it easy to determine the sign of the function on different intervals.

What do students get wrong about Quadratic Factored Form?

Not every quadratic factors nicely over the integers—some have irrational or complex roots.

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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