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Zeros and discriminant

The zeros of a function are where the graph crosses the x-axis, i.e. the solutions of $f(x)=0$ . For a quadratic function the sign of the discriminant $D$ tells us how many zeros the graph has.

D>0:\ \text{two zeros}

the parabola crosses the x-axis twice

D=0:\ \text{one zero}

the parabola touches the x-axis

D<0:\ \text{no zeros}

the parabola never reaches the x-axis

Symbols

D

discriminant,

D=b^2-4ac

Example

$f(x)=x^2-4x+4$ : $D=16-16=0$ , one zero $x=2$ .

$f(x)=x^2+2x+5$ : $D=4-20=-16<0$ , no zeros.

A vertex above the x-axis for a downward parabola gives no zeros - always check the sign of

a

together with

D

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Part of Foundations of Mathematics: Functions and graphs.