Sum of Roots of Quadratic Equation

Theorem

Let $P$ be the quadratic equation $a x^2 + b x + c = 0$.

Let $\alpha$ and $\beta$ be the roots of $P$.

Then:

$\alpha + \beta = -\dfrac b a$

$\ds \alpha$

$=$

$\ds \frac {-b + \sqrt {b^2 - 4 a c} } {2 a}$

$\ds \beta$

$=$

$\ds \frac {-b - \sqrt {b^2 - 4 a c} } {2 a}$

Without loss of generality, selecting $\alpha$ and $\beta$ as such

$\ds \leadsto \ \ $

$\ds \alpha + \beta$

$=$

$\ds \frac {-b - \sqrt {b^2 - 4 a c} - b + \sqrt {b^2 - 4 a c} } {2 a}$

$\ds $

$=$

$\ds \frac {-2 b} {2 a}$

$\ds $

$=$

$\ds -\frac b a$

$\blacksquare$

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: $a x^2 + b x + c = 0$: $9.2$
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: $a x^2 + b x + c = 0$: $5.2.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): quadratic equation