Solution to Quadratic Equation

Theorem

The quadratic equation of the form $a x^2 + b x + c = 0$ has solutions:

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

Real Coefficients

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.

Proof

$\ds a x^2 + b x + c$

$=$

$\ds 0$

by hypothesis

$\ds \leadsto \ \ $

$\ds \dfrac {\paren {2 a x + b}^2 - b^2 + 4 a c} {4 a}$

$=$

$\ds 0$

Completing the Square

$\ds \leadsto \ \ $

$\ds \paren {2 a x + b}^2$

$=$

$\ds b^2 - 4 a c$

simplification

$\ds \leadsto \ \ $

$\ds x$

$=$

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

solving for $x$

$\blacksquare$

Also known as

is often referred to as the quadratic formula.

Sources

1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): Chapter $\text I$ Introductory: $2$. The Use of Symbols
1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: $(1.20)$
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
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Polynomial Equations: $31$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quadratic formula
2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Where to begin...
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quadratic formula
2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $4$: Lure of the Unknown: Equations
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