Quadratic Equation

Theorem

An algebraic equation of the form $ax^2 + bx + c = 0$ is called a quadratic equation.

It has solutions $\displaystyle x = \frac {-b \pm \sqrt {b^2 - 4 a c}} {2a}$.

Discriminant

The expression $b^2 - 4 a c$ is called the discriminant of the equation.

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

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

Note that this is a special case of the general discriminant, although it is important to note that the general formula is given for monic polynomials.

Direct Proof

Let $ax^2 + bx + c = 0$. Then:

	\(\displaystyle \)	\(\displaystyle 4 a^2 x^2 + 4 a b x + 4 a c\)	\(=\)	\(\displaystyle 0\)	\(\displaystyle \)	(multiplying through by $4 a$)
	\(\displaystyle \implies\)	\(\displaystyle \left({2 a x + b}\right)^2 - b^2 + 4 a c\)	\(=\)	\(\displaystyle 0\)	\(\displaystyle \)	(Completing the square)
	\(\displaystyle \implies\)	\(\displaystyle \left({2 a x + b}\right)^2\)	\(=\)	\(\displaystyle b^2 - 4 a c\)	\(\displaystyle \)
	\(\displaystyle \implies\)	\(\displaystyle x\)	\(=\)	\(\displaystyle \frac {-b \pm \sqrt {b^2 - 4 a c} }{2a}\)	\(\displaystyle \)

• 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 $\displaystyle x = \frac {-b} {2 a}$.

• If the discriminant $b^2 - 4 a c < 0$, then we can write it as:

$b^2 - 4 a c = \left({-1}\right) \left|{b^2 - 4 a c}\right|$

Thus $\sqrt {b^2 - 4 a c} = \pm i \sqrt {\left|{b^2 - 4 a c}\right|}$, and the two solutions are:

$\displaystyle x = \frac {-b} {2 a} + i \frac {\sqrt {\left|{b^2 - 4 a c}\right|}} {2 a}, x = \frac {-b} {2 a} - i \frac {\sqrt {\left|{b^2 - 4 a c}\right|}} {2 a}$

and once again the result follows.

$\blacksquare$

Notes

Some older treatments of this subject report this as:

An algebraic eqn of the form $ax^2 + 2bx + c = 0$ is called a quadratic equation.

It has solutions $\displaystyle x = \frac {-b \pm \sqrt {b^2 - a c}} a$.

but this has fallen out of fashion.

Sources

• Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $9.1$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.10$