Math FoundationsEquationsQuadratic Equations

Factoring quadratics

Rewriting quadratic trinomials and quadratic equations as a product of factors instead of a sum of terms. Lets you solve equations quickly, simplify messy expressions, and spot the roots immediately.
Key elements:
Connections:
Completing the square is required to factor a quadratic trinomial in the general case, so you absolutely need to know how to use this method.
Adding and compensating
Factoring a quadratic trinomial uses the discriminant and the general root formula for quadratic equations. You need all of that, no excuses.
The quadratic formula
Discriminant
Biquadratic equation
Statistics:
Statement1
Important1
Problem9
Article
Summary
Practice
Possible factorization cases for a quadratic trinomial

Why factor at all?

  • 1
    You can simplify expressions
    Writing a quadratic trinomial as a product of factors often lets you simplify complicated expressions:
    5⋅(x2−2x−24)(x+4)⋅(x−6)⋅10=5⋅(x+4)⋅(x−6)10⋅(x+4)⋅(x−6)=510=12=0.5\frac{5 \cdot (x^2 - 2x - 24)}{(x+4) \cdot (x-6) \cdot 10} = \frac{5 \cdot \cancel{(x+4)} \cdot \cancel{(x-6)}}{10 \cdot \cancel{(x+4)} \cdot \cancel{(x-6)}} = \frac{5}{10} = \frac{1}{2} = 0.5
  • 2
    Getting back to the sum is easy
    Once a quadratic trinomial is factored, getting the sum form back is just a matter of expanding the brackets:
    2(x−1)(4+x)=2(4x+x2−4−x)=2(x2+3x−4)=2x2+6x−82(x-1)(4+x) = 2(4x + x^2 - 4 -x) = 2(x^2 + 3x - 4) = \boxed{2x^2 + 6x - 8}
    But going the other way, from the sum form to factors, is much harder!
    2x2+6x−8=… ? …=2(x−1)(4+x)2x^2 + 6x - 8 = \ldots \text{ ? } \ldots = \boxed{2(x-1)(4+x)}
  • 3
    The roots become visible immediately
    Writing a quadratic trinomial as a product of factors lets you immediately see the roots of its “equation.” In that form it falls under Zero factors — you just set each factor equal to zero separately and get a true equality 0 = 0:
    In factor form, the roots are the numbers next to x, but with the opposite sign!
  • 4
    It's a way to solve quadratic equations
    Factoring is one more way to solve quadratic equations, alongside the methods you already know: completing the square and the general root formula. We rewrite the quadratic trinomial as multiplication and immediately see the roots if that expression were a quadratic equation. A big plus is that for simple quadratics, this lets you find the roots fast, sometimes right in your head!

    • Factoring by hand

    Simple quadratic trinomials can sometimes be factored by hand. To do that, you rewrite coefficient B as the sum of two numbers, and coefficient C as the product of those exact same two numbers. Geometrically, that means you take a few smaller shapes and assemble one big rectangle out of them.

    Examples of hand factoring

    👀
    Example
    Factor the quadratic trinomial and find the roots of the corresponding quadratic equation:
    x2+5x+6x^2 + 5x + 6

    Factoring in the general case

    Factoring a quadratic trinomial

    If a quadratic trinomial has roots (let's call them x1x_1 and x2x_2), then that trinomial can always be factored:
    These are two different notations, one through addition and one through multiplication, but they mean the same thing, just like 10 + 6 and 2⋅82\cdot8 denote the same number. You can switch a quadratic trinomial into factor form and back again in absolutely any situation!

    Examples of factoring through roots

    👀
    Example
    Factor the quadratic trinomial by solving the corresponding quadratic equation:
    4x2+15x−44x^2 + 15x - 4
    Factoring quadratics