Solution to Quadratic Equation/Real Coefficients

Theorem

Let $a, b, c \in \R$.

The quadratic equation $a x^2 + b x + c = 0$ has:

Two real solutions if $b^2 - 4 a c > 0$

One real solution if $b^2 - 4 a c = 0$

Two complex solutions if $b^2 - 4 a c < 0$, and those two solutions are complex conjugates.

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

If the discriminant $b^2 - 4 a c > 0$ then $\sqrt {b^2 - 4 a c}$ has two values and the result follows.

If the discriminant $b^2 - 4 a c = 0$ then $\sqrt {b^2 - 4 a c} = 0$ and $x = \dfrac {-b} {2 a}$.

If the discriminant $b^2 - 4 a c < 0$, then it can be written as:

$b^2 - 4 a c = \paren {-1} \size {b^2 - 4 a c}$

Thus:

$\sqrt {b^2 - 4 a c} = \pm i \sqrt {\size {b^2 - 4 a c} }$

and the two solutions are:

$x = \dfrac {-b} {2 a} + i \dfrac {\sqrt {\size {b^2 - 4 a c} } } {2 a}, x = \dfrac {-b} {2 a} - i \dfrac {\sqrt {\size {b^2 - 4 a c} } } {2 a}$

and once again the result follows.

$\blacksquare$

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.8$ Algebraic Equations: Solution of Quadratic Equations: $3.8.1$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 9$: Solutions of Algebraic Equations: Quadratic Equation: $a x^2 + b x + c = 0$: $9.1$
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.10$: Quadratic equations
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): discriminant (of a polynomial equation)
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): discriminant (of a polynomial equation)
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 5$: Solutions of Algebraic Equations: Quadratic Equation: $a x^2 + b x + c = 0$: $5.1.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): quadratic equation