Factored form and relationships between roots

Factored form

When the discriminant delta is positive or zero, the trinomial ax^2+bx+c can be written in factored form as:

a*(x - x₁)*(x - x₂) = 0

where x₁ and x₂ are the roots found using the discriminant. This form is very useful because a product of factors is zero if and only if one of the factors is zero.

Example

Let’s take x² - 5x + 6 = 0, whose roots are x₁ = 2 and x₂ = 3. The factored form is: (x - 2)(x - 3) = 0

Relations between coefficients and roots (Vieta’s formula)

Even without calculating delta, we can relate the roots to the coefficients:

sum S = x₁ + x₂ = -b/a product P = x₁ * x₂ = c/a

Verification using the example: S = 2 + 3 = 5 = -(-5)/1. P = 2 * 3 = 6 = 6/1. This matches.

Practical use

These relationships allow us to quickly check a result, or to find one root if we already know the other: if x₁ is known, x₂ = S – x₁ = –b/a – x₁.

Common pitfall

Do not confuse the sum and the product: S = -b/a (with a minus sign), P = c/a (without a minus sign). Mixing up these formulas is a common mistake.