# Quadratic functions, roots and discriminant

> IB Mathematics: Analysis and Approaches SL · Functions
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u2-quadratic-functions-roots-and-discriminant/

This subtopic explains how to use the discriminant of a quadratic function to find the number and nature of its roots, which correspond to x-intercepts of the parabola. We also apply the discriminant to solve common exam problems with unknown coefficients.

**Prerequisites:** [Quadratic function basics and graphs](https://www.owlsprep.com/study/ib-math-aa-sl-u2-introduction-to-quadratic-functions/); [Quadratic formula](https://www.owlsprep.com/study/ib-math-aa-sl-u2-quadratic-equations/)

## Learning objectives

- Calculate the discriminant of a quadratic equation
- Determine the number and nature of quadratic roots using the discriminant
- Relate discriminant values to the number of x-intercepts of a quadratic graph
- Solve exam problems involving unknown coefficients and discriminant conditions

## Roots and the discriminant: Key definition

**Discriminant** — For a quadratic equation in standard form $ax^2 + bx + c = 0$ where $a \neq 0$, the discriminant is the value calculated from coefficients that tells us the nature of the roots, with formula $\Delta = b^2 - 4ac$.

*Notation:* $\Delta$

*Example:* For $2x^2 - 5x + 1 = 0$, $\Delta = (-5)^2 - 4(2)(1) = 17$.

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

**Worked example:** Calculate the discriminant of the quadratic equation $3x^2 + 2x - 4 = 0$.

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

## Interpreting the discriminant: Nature of roots

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

> **warning**
>
> A repeated root is still one real root, not two. Examiners frequently deduct marks for incorrectly stating $\Delta=0$ gives two roots.

**Worked example:** The quadratic function is $f(x) = x^2 - 4x + 5$. 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. $$\Delta < 0 \implies \text{0 real roots, so 0 x-intercepts}$$

## Applying the discriminant to problems with unknown coefficients

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 $\Delta$ in terms of the unknown, apply the condition for $\Delta$, then solve the resulting equation or inequality.

**Worked example:** Find the range of values of $k$ for which $x^2 + 2kx + 9 = 0$ 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 $\Delta > 0$:
6. $$4k^2 - 36 > 0 \\ k^2 - 9 > 0 \\ (k - 3)(k + 3) > 0$$
7. Solve the inequality to get the final solution:
8. $$k < -3 \text{ or } k > 3$$

**Check your understanding**

Check your understanding:

1. What is the correct condition for the parabola $y = ax^2 + bx + c$ (with $a>0$) to always lie above the x-axis?

   - $\Delta < 0$
   - $\Delta = 0$
   - $\Delta > 0$
   - $a > 0$

   *Why:* Correct! If $a>0$ the parabola opens upwards, and if $\Delta < 0$ it never crosses the x-axis, so it is always above the x-axis.

## Common pitfalls

- **Wrong:** Taking c with the wrong sign after rearranging the quadratic to equal zero
  - Why it fails: Sign errors are the most common mistake in discriminant calculations. A negative c flips the sign of the $4ac$ term.
  - Correct: Always rearrange the equation to $ax^2 + bx + c = 0$ first, then write down a, b, c including their explicit signs before calculating $\Delta$.
- **Wrong:** Stating $\Delta = 0$ gives two real roots
  - Why it fails: $\Delta = 0$ gives one repeated (equal) root, not two distinct roots. This is a common marked error in exams.
  - Correct: Use the phrasing "one repeated real root" or "one real root" when $\Delta = 0$.
- **Wrong:** Writing $\Delta \geq 0$ for two distinct roots, or $\Delta > 0$ for at least one real root
  - Why it fails: Wording of the question directly defines the required inequality. Mismatching the inequality loses easy marks.
  - Correct: Match the condition exactly: two distinct roots → $\Delta > 0$, at least one real root → $\Delta \geq 0$.
- **Wrong:** Forgetting to check the leading coefficient is not zero when it is unknown
  - Why it fails: 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: Always confirm $a \neq 0$ first when the leading coefficient contains an unknown constant.
- **Wrong:** Miscalculating $-4ac$ when c is negative: e.g. $c=-2$, writing $-4a(-2) = -8a$
  - Why it fails: Double negative multiplication errors are extremely common in discriminant calculations.
  - Correct: Double-check the sign of the $-4ac$ term: negative × negative = positive.

## Cheatsheet

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

## 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.

- [Reciprocal and rational functions](https://www.owlsprep.com/study/ib-math-aa-sl-u2-reciprocal-and-rational-functions/)
- [Exponential and logarithmic functions and graphs](https://www.owlsprep.com/study/ib-math-aa-sl-u2-exponential-and-logarithmic-functions-and/)
- [Geometry & Trigonometry](https://www.owlsprep.com/study/ib-math-aa-sl-u3-overview/)

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