Zero products and quadratic equations

A zero product gives alternatives

For 2x²-5x-3 = 0, factor to (2x + 1)(x-3) = 0. A product of real factors is zero only when at least one factor is zero. Thus x = -1/2 or x = 3. Check both in the original.

Do not apply this rule to (2x + 1)(x-3) = 6: the product is not zero. Bring all terms to one side before choosing a method.

Expand or factor only after making the right side zero

For x² + x = 6, first obtain x² + x − 6 = 0. Then (x + 3)(x − 2) = 0 gives x = −3 or 2. Setting x(x + 1) = 6 and making each factor zero would change the original condition.

Check your understanding 1 Do not divide away a root

Solve x² = 5x.

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Bring all terms to one side: x(x − 5) = 0, so x = 0 or 5. Dividing the original by x would lose the zero solution because that division is invalid at x = 0.

A context chooses allowed roots afterward

If a quadratic models a physical length, first solve the equation completely, then test each root against the original conditions. A negative value is rejected only when the model’s quantity or domain excludes it.