Quadratic equations

1 Quadratic equations

A quadratic equation is an equation with the form $a x^{2} + b x + c = 0$ where $x$ represents an unknown and $a$ , $b$ and $c$ are known numbers with $a \neq 0$ .

1.1 Solutions to a quadratic equation

A solution to a quadratic equation is a value of $x$ such that the equation balances. The solutions to quadratic equations can be found by using the quadratic formula:

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} .

(1)

Example.

For instance, the solutions to $x^{2} + 2 x - 3 = 0$ are:

\begin{array}{l} x & = \frac{- 2 \pm \sqrt{2^{2} - 4 \times 1 \times - 3}}{2 \times 1} \\ = \frac{- 2 \pm \sqrt{4 + 12}}{2} \\ = \frac{- 2 \pm \sqrt{16}}{2} \\ = \frac{- 2 \pm 4}{2} \end{array}

Hence, $x = 1$ or $x = - 3$ .

1.2 The discriminant

Definition (Discriminant).

The discriminant of a quadratic equation with coefficients $a, b, c \in ℝ$ is:

Δ = b^{2} - 4 a c .

Remark.

Note that this is the expression beneath the square root symbol in the quadratic formula (1).

We can use the discriminant to determine the number of real roots of a quadratic equation. The number depends on the value of $Δ$ as in table 1.


Value of $Δ$	Real roots

$Δ > 0$	Two, distinct
$Δ = 0$	One, repeated
$Δ < 0$	Zero

Table 1: Number of real roots of a quadratic equation, given the discriminant

Figure 1 shows an example of each possibility¹ .

Figure 1: Examples of quadratic functions with zero, one and two real roots.

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