The discriminant and the nature of roots

By the end of this lesson

By the end you can find the discriminant of $ax^2 + bx + c$ and use it to say how many real roots $ax^2 + bx + c = 0$ has, including when there's a repeated root.

The bit under the square root

Look at the formula again:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Everything hinges on $b^2 - 4ac$ :

Positive: $\pm$ a positive square root gives two different answers.
Zero: $\pm 0$ changes nothing, so both "answers" are the same.
Negative: there's no real square root, so no real roots at all.

Definition

Discriminant

The discriminant of the quadratic $ax^2 + bx + c$ (where $a \neq 0$ ) is

$b^2 - 4ac$

Its sign (positive, zero or negative) decides the nature of the roots of $ax^2 + bx + c = 0$ .

"Nature of the roots" is exam-speak for "how many real roots, and are they different or the same?"

Three cases

The sign of $b^2 - 4ac$ tells you which of three pictures you get:

$b^2 - 4ac$	Roots of $ax^2 + bx + c = 0$	Graph of $y = ax^2 + bx + c$
$> 0$	two distinct real roots	crosses the $x$ -axis twice
$= 0$	one repeated root (two equal real roots)	touches the $x$ -axis at its vertex
$< 0$	no real roots	doesn't meet the $x$ -axis

Definition

Repeated root

When $b^2 - 4ac = 0$ , the two roots land on the same number.

That number is called a repeated root. Questions also say equal roots.

The quadratic is then a perfect square:

$ax^2 + bx + c = a(x - \alpha)^2$

$\alpha$ (the Greek letter "alpha") is just a name for the repeated root.
From the formula with $\sqrt{0} = 0$ , it is $\alpha = -\dfrac{b}{2a}$ .

Example: $4x^2 - 12x + 9 = (2x - 3)^2$ . Setting $2x - 3 = 0$ gives the repeated root $x = \tfrac32$ .

Check with the formula: $-\dfrac{b}{2a} = -\dfrac{-12}{8} = \tfrac32$ ✓

Quick check

DefinitionQuestion 1 · Recall

Fill in the blanksQuestion 2 · Recall

Answer the 2 questions above to continue.