Math 2.0

Field theory and polynomials Galois theory Theorem

Quadratic formula

Builds on The complex numbers form a fieldExistence of nth roots of complex numbers

Statement

Let aa, bb and cc be complex numbers with a≠0a \neq 0, and let ss be a complex number with s2=b2−4acs^2 = b^2 - 4 a c. Then, for each complex number xx, ax2+bx+c=0  ⟺  x=−b+s2a or x=−b−s2a.a x^2 + b x + c = 0 \iff x = \frac{-b + s}{2 a} \text{ or } x = \frac{-b - s}{2 a}.

Remarks

Such an ss exists, by the existence of nth roots for n=2n = 2. The number Δ=b2−4ac\Delta = b^2 - 4 a c is the discriminant. For real a,b,ca, b, c, there are two real roots if Δ>0\Delta > 0, one double real root if Δ=0\Delta = 0, and two conjugate non-real roots if Δ<0\Delta < 0. The same formula holds in each field in which 2≠02 \neq 0.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionEuler's formula, statement, no evidence levelEuler's formulastatementProduct of exponentials, statement, no evidence levelProduct of exponentia…statementReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionRoot of unity, definition, no evidence levelRoot of unitydefinitionThe reals form a field, statement, no evidence levelThe reals form a fieldstatementSupremum, definition, no evidence levelSupremumdefinitionde Moivre's formula, statement, no evidence levelde Moivre's formulastatementThe complex numbers form a field, statement, no evidence levelThe complex numbers f…statementCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionArgument of a complex number, definition, no evidence levelArgument of a complex…definitionPrincipal value of the argument, statement, no evidence levelPrincipal value of th…statementPolar form, statement, no evidence levelPolar formstatementThe nth roots of unity, statement, no evidence levelThe nth roots of unitystatementNumber of nth roots of unity, statement, no evidence levelNumber of nth roots o…statementExistence of nth roots of complex numbers, statement, no evidence levelExistence of nth root…statementQuadratic formula, statement, no evidence levelQuadratic formulastatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Euler's formula statement
  5. Product of exponentials statement
  6. Real and imaginary parts definition
  7. Root of unity definition
  8. The reals form a field statement
  9. Supremum definition
  10. de Moivre's formula statement
  11. The complex numbers form a field statement
  12. Completeness axiom statement
  13. Existence of square roots of nonnegative reals statement
  14. Square root definition
  15. Modulus of a complex number definition
  16. Argument of a complex number definition
  17. Principal value of the argument statement
  18. Polar form statement
  19. The nth roots of unity statement
  20. Number of nth roots of unity statement
  21. Existence of nth roots of complex numbers statement
  22. Quadratic formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Quadratic_formula, lead section
  2. Web https://en.wikipedia.org/wiki/Quadratic_formula, Derivation by completing the square
About this page
Tag0KYZXYDGQZ
URIm2://0KYZXYDGQZ
Kindstatement, theorem
WritingW1 stated
MSC12D10
StatusThis copy of the site has no build data. Run m2 status 0KYZXYDGQZ.
UsesThe complex numbers form a field, Existence of nth roots of complex numbers
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