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↑ Note:
In 1751, Leonhard Euler proved that any body has a principal axis of rotation: Leonhard Euler (presented: October 1751; published: 1760) "Du mouvement d'un corps solide quelconque lorsqu'il tourne autour d'un axe mobile" (On the movement of any solid body while it rotates around a moving axis), Histoire de l'Académie royale des sciences et des belles lettres de Berlin, pp. 176–227.  On p. 212, Euler proves that any body contains a principal axis of rotation: "Théorem.  44.  De quelque figure que soit le corps, on y peut toujours assigner un tel axe, qui passe par son centre de gravité, autour duquel le corps peut tourner librement & d'un mouvement uniforme."  (Theorem.  44.  Whatever be the shape of the body, one can always assign to it such an axis, which passes through its center of gravity, around which it can rotate freely and with a uniform motion.)
In 1755, Johann Andreas Segner proved that any body has three principal axes of rotation: Johann Andreas Segner, Specimen theoriae turbinum [Essay on the theory of tops (i.e., rotating bodies)] (Halle ("Halae"), (Germany): Gebauer, 1755).  (https://books.google.com/books?id=29 p. xxviiii [29]), Segner derives a third-degree equation in t, which proves that a body has three principal axes of rotation.  He then states (on the same page): "Non autem repugnat tres esse eiusmodi positiones plani HM, quia in aequatione cubica radices tres esse possunt, et tres tangentis t valores."  (However, it is not inconsistent [that there] be three such positions of the plane HM, because in cubic equations, [there] can be three roots, and three values of the tangent t.)
The relevant passage of Segner's work was discussed briefly by Arthur Cayley.  See: A. Cayley (1862) "Report on the progress of the solution of certain special problems of dynamics," Report of the Thirty-second meeting of the British Association for the Advancement of Science; held at Cambridge in October 1862, 32: 184–252; see especially pp. 225–226.
↑  Kline 1972, pp. 807–808 Augustin Cauchy (1839) "Mémoire sur l'intégration des équations linéaires" (Memoir on the integration of linear equations), Comptes rendus, 8: 827–830, 845–865, 889–907, 931–937.  From p. 827: "On sait d'ailleurs qu'en suivant la méthode de Lagrange, on obtient pour valeur générale de la variable prinicipale une fonction dans laquelle entrent avec la variable principale les racines d'une certaine équation que j'appellerai l'équation caractéristique, le degré de cette équation étant précisément l'order de l'équation différentielle qu'il s'agit d'intégrer."  (One knows, moreover, that by following Lagrange's method, one obtains for the general value of the principal variable a function in which there appear, together with the principal variable, the roots of a certain equation that I will call the "characteristic equation", the degree of this equation being precisely the order of the differential equation that must be integrated.)
↑ See:
David Hilbert (1904) "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen.  (Erste Mitteilung)" (Fundamentals of a general theory of linear integral equations. (First report)), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (News of the Philosophical Society at Göttingen, mathematical-physical section), pp. 49–91.  From p. 51: "Insbesondere in dieser ersten Mitteilung gelange ich zu Formeln, die die Entwickelung einer willkürlichen Funktion nach gewissen ausgezeichneten Funktionen, die ich 'Eigenfunktionen' nenne, liefern: …" (In particular, in this first report I arrive at formulas that provide the [series] development of an arbitrary function in terms of some distinctive functions, which I call eigenfunctions: … )  Later on the same page: "Dieser Erfolg ist wesentlich durch den Umstand bedingt, daß ich nicht, wie es bisher geschah, in erster Linie auf den Beweis für die Existenz der Eigenwerte ausgehe, … " (This success is mainly attributable to the fact that I do not, as it has happened until now, first of all aim at a proof of the existence of eigenvalues, … )
For the origin and evolution of the terms eigenvalue, characteristic value, etc., see: Earliest Known Uses of Some of the Words of Mathematics (E)
↑  Determine the characteristic polynomial of A:
 {\begin{aligned}|A-\lambda I|&={\begin{vmatrix}a-\lambda &b\\c&d-\lambda \end{vmatrix}}\\&=(a-\lambda )(d-\lambda )-bc\\&=ad-a\lambda -d\lambda +\lambda ^{2}-bc\\&=\lambda ^{2}-(a+d)\lambda +(ad-bc)\\\end{aligned}}

Use the quadratic formula to find the values of λ:
 {\begin{aligned}\lambda &={\frac {(a+d)\pm {\sqrt {(a+d)^{2}-4(ad-bc)}}}{2}}\\&={\frac {a+d}{2}}\pm {\frac {\sqrt {(a+d)^{2}-4(ad-bc)}}{2}}\end{aligned}}

Simplify radicand:
 {\begin{aligned}&(a+d)^{2}-4(ad-bc)\\={}&a^{2}+2ad+d^{2}-4ad+4bc\\={}&a^{2}-2ad+d^{2}+4bc\\={}&(a-d)^{2}+4bc\end{aligned}}

Bring denominator into the square root:
 {\begin{aligned}\lambda &={\frac {a+d}{2}}\pm {\sqrt {\frac {(a-d)^{2}+4bc}{4}}}\\&={\frac {a+d}{2}}\pm {\sqrt {{\frac {1}{4}}(a-d)^{2}+bc}}\end{aligned}}
↑  For a proof of this lemma, see Roman 2008, Theorem 8.2 on p. 186; Shilov 1977, p. 109; Hefferon 2001, p. 364; Beezer 2006, Theorem EDELI on p. 469; and Lemma for linear independence of eigenvectors