Use factors to solve a quadratic equation

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Use factors to solve a quadratic equation

First rearrange a quadratic equation so one side is zero. A product is zero when at least one factor is zero. For x² − 4x + 3 = 0, the factorisation (x − 1)(x − 3) = 0 gives two possible inputs.

Keep both roots unless a stated context excludes one. Dividing an equation by an expression involving x can lose a root if that expression could be zero. The graph checks the interpretation: roots are the inputs where the curve crosses or touches the horizontal axis.

A graph can give approximate roots when factorisation is not convenient. A tangent contact with the axis represents a repeated root, while a parabola that stays entirely above or below the axis has no real roots. Reading precision depends on the scale.

Worked example: Solve x² − 4x + 3 = 0.

Factorise as (x − 1)(x − 3) = 0.

Set each factor equal to zero.

Substitute both values into the original equation.

Answer: x = 1 or x = 3.

Check your understanding: Solve x(x − 5) = 0.

Answer: x = 0 or x = 5.

Why: Dividing by x would incorrectly discard the zero root.