Zero product equation: A × B = 0 in 2 seconds

The rule

A product of factors is zero if and only if at least one factor is zero.

$A \times B = 0 \iff A = 0 \text{ or } B = 0$

Generalization: $A \times B \times C = 0 \iff A = 0$ or $B = 0$ or $C = 0$.

3-step method

1. Factor the expression until you get a product equal to 0.
2. Apply the rule: each factor may be zero.
3. Solve each simple equation, then collect the solutions.

Typical example

Solve $(x - 3)(2x + 1) = 0$.

• $x - 3 = 0 \Rightarrow x = 3$
• $2x + 1 = 0 \Rightarrow x = -1/2$

Solutions: $S = \{-1/2 \, , \, 3\}$.

Example needing factoring

Solve $x^2 - 5x = 0$.

Factor: $x(x - 5) = 0$.

• $x = 0$
• $x - 5 = 0 \Rightarrow x = 5$

$S = \{0 \, , \, 5\}$.

ONLY applies when the product equals 0

$A \times B = 12$ does NOT imply that $A = 12$ or $B = 12$ (you might have $A = 3, B = 4$).

⚠️ Common error: moving $12$ to the left but not factoring. Ex: $x(x-1) = 6$ does not give $x = 6$ or $x - 1 = 6$.

Common mistakes

1. Using the rule without a product (no $=0$ or not factored).
2. Missing a solution (not solving each factor).
3. Writing $x^2 = 25 \Rightarrow x = 5$ forgetting $-5$ (here $x^2 - 25 = 0$ ⇒ $(x-5)(x+5)=0$).

?À toi de jouer

Solutions of (2x − 4)(x + 3) = 0?

A.x = 2 or x = −3B.x = 4 or x = 3C.x = −2 or x = 3D.x = 4 or x = −3

Valider