Discriminant

Summary: The discriminant of a quadratic equation is the quantity under the square root in the quadratic formula; its sign determines whether the two roots are real and distinct, real and equal, or imaginary (impossible).

Sources: chapter-1.4.9

Last updated: 2026-05-03

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Definition

For the quadratic $x^2-ax+b=0$, the solution is:

$x=\frac{a}{2}\pm \sqrt{\frac{a^2}{4}-b}$

The discriminant is $\Delta =\frac{a^2}{4}-b$ (equivalently $\Delta =a^2-4b$ for the unscaled form). Its sign controls the nature of the roots (source: chapter-1.4.9, §701).

Three cases

Condition	$\Delta$	Roots
$b<\frac{1}{4}{a}^2$	$\Delta >0$	Two distinct real roots
$b=\frac{1}{4}{a}^2$	$\Delta =0$	One repeated real root ($x=a/2$)
$b>\frac{1}{4}{a}^2$	$\Delta <0$	Two imaginary roots; the equation is impossible

General form (§702)

For $f{x}^2-gx+h=0$ the roots are:

$x=\frac{g\pm \sqrt{g^2-4fh}}{2f}$

The roots are imaginary when $4fh>g^2$, i.e., when four times the product of the leading and constant coefficients exceeds the square of the linear coefficient.

Imaginary roots are not approximable

Euler stresses (§702) that irrational real roots can always be approached by approximation (see square-roots-and-irrational-numbers), but imaginary roots like $\sqrt{-5}$ admit no approximation whatsoever — no real number is close to an imaginary one. This sharpens the distinction between irrational and imaginary quantities.

Vieta relations still hold

Even when $\Delta <0$ and the roots are imaginary, their sum equals $a$ and their product equals $b$ (see vieta-formulas). The roots are always expressible, just not always real.

Related pages

• quadratic-equations
• vieta-formulas
• completing-the-square
• imaginary-numbers
• ch1.4.9-nature-quadratic-equations
• ch1.4.5-pure-quadratic-equations