Factor a trinomial when you know its roots

The formula to remember

If the trinomial $ax^2 + bx + c$ has two roots $x_1$ and $x_2$, then:

$ax^2 + bx + c = a(x - x_1)(x - x_2)$

Never forget the factor $a$ in front.

Example

$3x^2 - 12x + 9$:

• Discriminant: $\Delta = 144 - 108 = 36 > 0$.
• Roots: $x_1 = \dfrac{12 - 6}{6} = 1$, $x_2 = \dfrac{12 + 6}{6} = 3$.
• Factorization: $3(x - 1)(x - 3)$.

Check by expansion: $3(x-1)(x-3) = 3(x^2 - 4x + 3) = 3x^2 - 12x + 9$ ✓.

Case $\Delta = 0$: double root

If the trinomial has a double root $x_0$:

$ax^2 + bx + c = a(x - x_0)^2$

Case $\Delta < 0$: impossible

If $\Delta < 0$, the trinomial doesn’t factor over the reals. It stays in expanded form (or factors with complex numbers in advanced classes).

Quick check

After factoring, mentally expand only the constant terms to verify it equals $c$.

On the example: $3 \times (-1) \times (-3) = 9$ ✓.

?Your turn

How does x² − 7x + 12 factor?

A.(x − 3)(x − 4)B.(x + 3)(x + 4)C.(x − 3)(x + 4)D.(x − 7)(x + 12)

Check