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Factored Form

Factored form is a way of writing a quadratic expression as a product of its leading coefficient and two binomial factors: a(x−r1)(x−r2)a(x - r_1)(x - r_2). The values r1r_1 and r2r_2 are the roots (or zeros) of the quadratic, meaning they are the xx-values where the graph crosses the xx-axis.

A quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c is in factored form when it is expressed as f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2), where a≠0a \neq 0 and r1r_1, r2r_2 are the roots of the equation ax2+bx+c=0ax^2 + bx + c = 0. The constant aa is the same leading coefficient that appears in standard form and controls the parabola's width and direction. Every quadratic with real roots can be written in factored form, and doing so makes the roots immediately visible without further calculation.

Key Formula

f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2)
Where:
  • aa = the leading coefficient (same as in standard form)
  • r1r_1 = the first root (x-intercept) of the quadratic
  • r2r_2 = the second root (x-intercept) of the quadratic

Worked Example

Problem: Write f(x) = 2x² − 12x + 16 in factored form and identify the roots.
Step 1: Factor out the leading coefficient from all three terms.
f(x)=2(x2−6x+8)f(x) = 2(x^2 - 6x + 8)
Step 2: Factor the trinomial inside the parentheses. Find two numbers that multiply to 8 and add to −6. Those numbers are −2 and −4.
x2−6x+8=(x−2)(x−4)x^2 - 6x + 8 = (x - 2)(x - 4)
Step 3: Write the complete factored form by including the leading coefficient.
f(x)=2(x−2)(x−4)f(x) = 2(x - 2)(x - 4)
Step 4: Read the roots directly from the factors. Set each factor equal to zero: x − 2 = 0 gives x = 2, and x − 4 = 0 gives x = 4.
r1=2,r2=4r_1 = 2, \quad r_2 = 4
Answer: The factored form is f(x)=2(x−2)(x−4)f(x) = 2(x - 2)(x - 4), with roots at x=2x = 2 and x=4x = 4.

Visualization

Why It Matters

Factored form is one of the fastest ways to find where a parabola crosses the xx-axis, which matters in physics (projectile motion), engineering, and optimization problems. When you graph a quadratic, knowing the roots immediately tells you two key points on the curve, and the axis of symmetry sits exactly halfway between them. Many real-world problems—like finding when a launched object hits the ground—reduce to reading roots from factored form.

Common Mistakes

Mistake: Forgetting the subtraction signs and writing (x + r₁)(x + r₂) when the roots are positive.
Correction: The formula uses minus signs: a(x − r₁)(x − r₂). If a root is 3, the factor is (x − 3), not (x + 3). A positive root produces a subtraction in the factor.
Mistake: Dropping the leading coefficient a when factoring.
Correction: The leading coefficient is part of the factored form. If standard form starts with 2x², the factored form must include that 2 out front: 2(x − r₁)(x − r₂). Leaving it out changes the function.

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