Quadratic functions, roots and discriminant

IB Mathematics: Analysis and Approaches SL· Unit 2: Functions, Topic 2.5· 15 min read

1. Roots and the discriminant: Key definition★★☆☆☆⏱ 5 min

📘 Definition

Discriminant

$Δ$

For a quadratic equation in standard form $a x^{2} + b x + c = 0$ where $a \neq = 0$ , the discriminant is the value calculated from coefficients that tells us the nature of the roots, with formula $Δ = b^{2} - 4 a c$ .

Example:

For $2 x^{2} - 5 x + 1 = 0$ , $Δ = (- 5)^{2} - 4 (2) (1) = 17$ .

Roots of the quadratic equation $a x^{2} + b x + c = 0$ are the x-values that satisfy the equation. These correspond exactly to the x-intercepts of the quadratic function $f (x) = a x^{2} + b x + c$ .

📐 Worked Example

Calculate the discriminant of the quadratic equation $3 x^{2} + 2 x - 4 = 0$ .

1
First, identify coefficients from the standard form $a x^{2} + b x + c = 0$ :
2
$a = 3, \quad b = 2, \quad c = -4$
3
Substitute into the discriminant formula to calculate $Δ$ :
4
$\Delta = (2)^2 - 4(3)(-4) = 4 + 48 = 52$

2. Interpreting the discriminant: Nature of roots★★☆☆☆⏱ 5 min

The value of the discriminant tells us how many real roots (and how many x-intercepts) the quadratic has. IB AA SL only works with real roots, so we have three distinct cases:

If $Δ > 0$ : Two distinct real roots → two distinct x-intercepts
If $Δ = 0$ : One repeated (equal) real root → parabola touches the x-axis at exactly one point (the vertex lies on the x-axis)
If $Δ < 0$ : No real roots → parabola never intersects the x-axis, all function values have the same sign as $a$

📐 Worked Example

The quadratic function is $f (x) = x^{2} - 4 x + 5$ . Determine how many x-intercepts it has.

1
To find x-intercepts, set $f (x) = 0$ to get the quadratic equation:
2
$x^2 - 4x + 5 = 0$
3
Calculate the discriminant:
4
$a = 1, \quad b = -4, \quad c = 5 \\ \Delta = (-4)^2 - 4(1)(5) = 16 - 20 = -4$
5
Interpret the result to get the answer:
6
$\Delta < 0 \implies \text{0 real roots, so 0 x-intercepts}$

3. Applying the discriminant to problems with unknown coefficients★★★☆☆⏱ 5 min

The most common exam question on this topic gives a quadratic with an unknown constant (usually called $k$ ) and a condition on the roots, then asks for possible values of $k$ . The method is always the same: write $Δ$ in terms of the unknown, apply the condition for $Δ$ , then solve the resulting equation or inequality.

📐 Worked Example

Find the range of values of $k$ for which $x^{2} + 2 k x + 9 = 0$ has two distinct real roots.

1
Identify coefficients in terms of $k$ :
2
$a = 1, \quad b = 2k, \quad c = 9$
3
Calculate the discriminant and simplify:
4
$\Delta = (2k)^2 - 4(1)(9) = 4k^2 - 36$
5
Apply the condition for two distinct real roots, which requires $Δ > 0$ :
6
$4k^2 - 36 > 0 \\ k^2 - 9 > 0 \\ (k - 3)(k + 3) > 0$
7
Solve the inequality to get the final solution:
8
$k < -3 \text{ or } k > 3$

✓ Quick check

Check your understanding:

What is the correct condition for the parabola $y = a x^{2} + b x + c$ (with $a > 0$ ) to always lie above the x-axis?
- $Δ < 0$
- $Δ = 0$
- $Δ > 0$
- $a > 0$
Reveal answer
$\Delta < 0$ —
Correct! If $a > 0$ the parabola opens upwards, and if $Δ < 0$ it never crosses the x-axis, so it is always above the x-axis.

4. Common Pitfalls

Wrong move:

Taking c with the wrong sign after rearranging the quadratic to equal zero

Why:

Sign errors are the most common mistake in discriminant calculations. A negative c flips the sign of the $4 a c$ term.

Correct move:

Always rearrange the equation to $a x^{2} + b x + c = 0$ first, then write down a, b, c including their explicit signs before calculating $Δ$ .

Wrong move:

Stating $Δ = 0$ gives two real roots

Why:

$Δ = 0$ gives one repeated (equal) root, not two distinct roots. This is a common marked error in exams.

Correct move:

Use the phrasing "one repeated real root" or "one real root" when $Δ = 0$ .

Wrong move:

Writing $Δ \geq 0$ for two distinct roots, or $Δ > 0$ for at least one real root

Why:

Wording of the question directly defines the required inequality. Mismatching the inequality loses easy marks.

Correct move:

Match the condition exactly: two distinct roots → $Δ > 0$ , at least one real root → $Δ \geq 0$ .

Wrong move:

Forgetting to check the leading coefficient is not zero when it is unknown

Why:

If the coefficient of $x^{2}$ is unknown, a value that makes it zero gives a linear equation, not a quadratic, changing the number of roots.

Correct move:

Always confirm $a \neq = 0$ first when the leading coefficient contains an unknown constant.

Wrong move:

Miscalculating $- 4 a c$ when c is negative: e.g. $c = - 2$ , writing $- 4 a (- 2) = - 8 a$

Why:

Double negative multiplication errors are extremely common in discriminant calculations.

Correct move:

Double-check the sign of the $- 4 a c$ term: negative × negative = positive.

5. Quick Reference Cheatsheet

Discriminant value	Number of real roots	Number of x-intercepts	Graph description
$Δ > 0$	2 distinct	2	Crosses x-axis at two points
$Δ = 0$	1 repeated	1	Touches x-axis at vertex
$Δ < 0$	0	0	Never crosses the x-axis

Going deeper

WebsiteKhan Academy: Discriminant review

What's Next

Mastering discriminant and roots of quadratics is a foundational skill that extends to almost all other algebra and geometry topics in IB AA SL. You will use these exact same ideas when finding intersections between lines and curves: you set the equations equal to get a quadratic, then use the discriminant to find how many intersection points exist. This is a very common question in coordinate geometry and tangent problems. You will also extend these concepts to higher-order polynomials when working with cubic and quartic functions later in the course. Building fluency with the discriminant now will help you avoid simple errors in these more complex problems.