Business Mathematics and Basic Statistics · Ch 1 — Quadratic Equations

The Discriminant and the Number of Real Roots

The quantity $D = b^2 - 4ac$ appearing under the square root in Sridharacharya's formula is called the discriminant of the quadratic equation. Its sign — without even computing the roots themselves — tells us the cardinality of the set of real roots, i.e. how many real roots the equation has:

Note

The Three Cases

$D > 0$ : the equation has two distinct real roots ( $x_1 \neq x_2$ ), since $\sqrt{D}$ is a positive real number and the $\pm$ sign genuinely produces two different values.
$D = 0$ : the equation has exactly one real root (a repeated or "equal" root, $x_1 = x_2 = -\dfrac{b}{2a}$ ), since $\sqrt{D} = 0$ collapses the $\pm$ into a single value.
$D < 0$ : the equation has no real root — $\sqrt{D}$ would require the square root of a negative number, which does not exist within the real numbers $\mathbb{R}$ . …

Definition 5Discriminant

$D = b^2 - 4ac$ for the quadratic $ax^2+bx+c=0$ . Its sign classifies the roots: $D>0$ gives two distinct real roots, $D=0$ gives one repeated real root …

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