Quadratic Equations

Summary: A quadratic equation is a polynomial equation of degree two; Euler classifies them as pure (no linear term) or mixt (complete), solves both by square-root extraction or completing the square, and establishes that every quadratic has exactly two roots.

Sources: chapter-1.4.5, chapter-1.4.6, chapter-1.4.9

Last updated: 2026-05-03

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Definition and general form

An equation of the second degree contains $x^2$ as its highest power. After collecting like terms and bringing all terms to one side, every quadratic takes the form (source: chapter-1.4.5, §626):

$a{x}^2\pm bx\pm c=0$

Pure vs. mixt quadratics

Pure (incomplete): the linear term $bx$ is absent.

$a{x}^2\pm c=0⟹x=\pm \sqrt{\frac{c}{a}}$

Solved directly by square-root extraction. See ch1.4.5-pure-quadratic-equations.

Mixt (complete): all three terms present. Requires completing-the-square or a substitution to reduce to a pure quadratic. See ch1.4.6-mixt-quadratic-equations.

The quadratic formula

For $a{x}^2+bx+c=0$, completing the square gives:

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

In Euler’s notation (with the equation written as $x^2=-px+q$):

$x=-\frac{1}{2}p\pm \sqrt{\frac{1}{4}{p}^2+q}$

Two solutions

Every quadratic has exactly two solutions (counting multiplicity). This follows from the factored form $(x-p)(x-q)=0$: the product is zero iff one factor is zero. See ch1.4.9-nature-quadratic-equations and vieta-formulas.

Nature of the roots (discriminant)

For $x^2-ax+b=0$, the discriminant is $\Delta =a^2-4b$:

| $\Delta >0$ | Two distinct real roots | |$\Delta =0$ | One repeated real root | |$\Delta <0$ | Two conjugate imaginary roots; problem is impossible |

See discriminant and imaginary-numbers.

Related pages

• completing-the-square
• vieta-formulas
• discriminant
• ch1.4.5-pure-quadratic-equations
• ch1.4.6-mixt-quadratic-equations
• ch1.4.9-nature-quadratic-equations
• equations
• imaginary-numbers