Quadratics¶
Recall (doc/postdoc). Not a first course.
Job: degree-\(2\) equations over fields; discriminant controls splitting; Galois group \(\le S_2\).
Core¶
\[
ax^2+bx+c=0\ (a\neq0),\qquad
\Delta=b^2-4ac,\qquad
x=\frac{-b\pm\sqrt{\Delta}}{2a}
\]
| \(\Delta\) over \(\mathbb{R}\) | Roots |
|---|---|
| \(>0\) | two distinct real |
| \(=0\) | double real |
| \(<0\) | conjugate complex pair |
Complete square: \(a(x+\frac{b}{2a})^2+\frac{4ac-b^2}{4a}=0\).
Vieta: sum \(=-b/a\), product \(=c/a\).
Methods: factor (if splits), square root form, complete square, quadratic formula (same, char \(\neq2\)).
Used later for¶
- Quadratic functions: geometry of \(f(x)=ax^2+bx+c\)
- Galois theory: quadratic extensions \(K(\sqrt{d})\)
- Conic sections: degenerate cases
- ODEs: characteristic \(r^2+br+c=0\)
Check¶
In char \(2\), why does the usual quadratic formula fail?
Division by \(2a\) and \(\Delta=b^2-4ac\) use \(2\neq0\); Artin–Schreier theory replaces for \(x^2+x+a\) forms.