The Discriminant of a Quadratic — CAIE A-Level Pure Mathematics 1 (9709) — Mathematics

Introduction

The discriminant is a single expression derived from the coefficients of a quadratic polynomial that instantly reveals how many real roots the corresponding equation has. In the 9709 exam, discriminant questions appear across a wide range of contexts: determining whether a curve intersects a line, proving a quadratic is always positive, and finding unknown constants given information about the nature of roots. Mastering the discriminant is therefore essential, not only as a standalone skill but as a tool woven throughout the Pure Mathematics 1 paper.

Core Concept

Recall from Completing the Square that the general quadratic equation $ax^2 + bx + c = 0$ (with $a \neq 0$ ) can be solved by the quadratic formula:

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

The expression under the square root, $b^2 - 4ac$ , controls whether real solutions exist and how many there are. This expression is called the discriminant.

If $b^2 - 4ac > 0$ : the square root gives two distinct real values, producing two distinct real roots.
If $b^2 - 4ac = 0$ : the square root vanishes, and the $\pm$ gives only one value, producing a repeated root (also called an equal root).
If $b^2 - 4ac < 0$ : the square root of a negative number has no real value, so there are no real roots.

Geometrically, these three cases describe how the parabola $y = ax^2 + bx + c$ relates to the $x$ -axis.

Key Formulae & Definitions

The Discriminant:

\Delta = b^2 - 4ac

where $ax^2 + bx + c$ is a quadratic polynomial with $a \neq 0$ .

Classification of roots:

Condition	Nature of roots
$\Delta > 0$	Two distinct real roots
$\Delta = 0$	One repeated (equal) root: $x = -\dfrac{b}{2a}$
$\Delta < 0$	No real roots

Note: A repeated root is a single value of $x$ at which the parabola just touches (is tangent to) the $x$ -axis.

Worked Examples

Example 1 — Determining the nature of roots

Determine the nature of the roots of $3x^2 - 5x + 4 = 0$ .

Step 1: Identify the coefficients.

a = 3, \quad b = -5, \quad c = 4

Step 2: Calculate the discriminant.

\Delta = b^2 - 4ac = (-5)^2 - 4(3)(4) = 25 - 48 = -23

Step 3: Interpret the result.

Since $\Delta = -23 < 0$ , the equation has no real roots. The parabola lies entirely above the $x$ -axis (as $a = 3 > 0$ ).

Example 2 — Finding an unknown constant given a repeated root

The equation $x^2 + kx + 9 = 0$ has a repeated root. Find the possible values of $k$ .

Step 1: Identify the coefficients.

a = 1, \quad b = k, \quad c = 9

Step 2: Set the discriminant equal to zero (condition for a repeated root).

\Delta = k^2 - 4(1)(9) = 0

k^2 - 36 = 0

Step 3: Solve for $k$ .

k^2 = 36 \implies k = \pm 6

Step 4: State the repeated roots for each case.

When $k = 6$ : repeated root at $x = -\dfrac{6}{2} = -3$ .
When $k = -6$ : repeated root at $x = -\dfrac{-6}{2} = 3$ .

Example 3 — Using the discriminant to find a range of values

Find the values of the constant $m$ for which $mx^2 + 6x + 3 = 0$ has two distinct real roots.

Step 1: Identify the coefficients (noting $m \neq 0$ for a quadratic).

a = m, \quad b = 6, \quad c = 3

Step 2: Write the condition for two distinct real roots.

\Delta > 0 \implies 36 - 12m > 0

Step 3: Solve the inequality.

36 > 12m \implies m < 3

Step 4: State the full constraint.

Combined with $m \neq 0$ : the equation has two distinct real roots when $m < 3$ and $m \neq 0$ .

Common Mistakes & Examiner Pitfalls

Sign errors with $b$ : The discriminant uses $b^2$ , which is always non-negative regardless of the sign of $b$ . A frequent error is computing $b^2$ incorrectly when $b$ is negative, e.g. writing $(-5)^2 = -25$ . Always square the full signed value.
Forgetting $a \neq 0$ : When $m$ (or another letter) is the leading coefficient, students often forget that $m = 0$ must be excluded separately, since the expression would no longer be quadratic.
Using $\Delta \geq 0$ instead of $\Delta > 0$ : "Two distinct real roots" requires strictly $\Delta > 0$ . The boundary $\Delta = 0$ gives a repeated root — a separate case. Read the question wording with care.
Confusing "no real roots" with "no roots": The discriminant only classifies real roots. In 9709 Pure 1, "no real roots" is the correct conclusion; do not write "no solutions" without this qualifier.
Not simplifying $\Delta$ before interpreting: In harder problems, $\Delta$ involves the unknown — students sometimes forget to rearrange the inequality correctly, particularly flipping the inequality sign when dividing by a negative value.

Practice Questions

Q1. Find the discriminant of $2x^2 - 7x + 5$ and state the nature of the roots of $2x^2 - 7x + 5 = 0$ .

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$a = 2,\ b = -7,\ c = 5$

\Delta = (-7)^2 - 4(2)(5) = 49 - 40 = 9

Since $\Delta = 9 > 0$ , the equation has two distinct real roots.

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Q2. The equation $4x^2 + px + 25 = 0$ has a repeated root. Find the possible values of $p$ and state the corresponding repeated root in each case.

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For a repeated root, $\Delta = 0$ :

p^2 - 4(4)(25) = 0 \implies p^2 = 400 \implies p = \pm 20

$p = 20$ : repeated root $x = -\dfrac{20}{8} = -\dfrac{5}{2}$
$p = -20$ : repeated root $x = -\dfrac{-20}{8} = \dfrac{5}{2}$

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Q3. Show that the equation $x^2 + 3x + 7 = 0$ has no real roots.

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$a = 1,\ b = 3,\ c = 7$

\Delta = 3^2 - 4(1)(7) = 9 - 28 = -19

Since $\Delta = -19 < 0$ , the equation has no real roots. $\square$

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Q4. Find the range of values of $k$ for which $x^2 + kx + k + 3 = 0$ has no real roots.

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For no real roots, $\Delta < 0$ :

k^2 - 4(1)(k + 3) < 0

k^2 - 4k - 12 < 0

(k - 6)(k + 2) < 0

The quadratic $(k-6)(k+2)$ is negative between its roots:

-2 < k < 6

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Q5. The line $y = 3x + c$ is a tangent to the curve $y = x^2 + 5x - 2$ . Find the value of $c$ .

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At a tangency, the line and curve intersect at exactly one point — a repeated root condition.

Set equal:

x^2 + 5x - 2 = 3x + c

x^2 + 2x - 2 - c = 0

For a repeated root, $\Delta = 0$ :

4 - 4(1)(-2 - c) = 0

4 + 8 + 4c = 0

4c = -12 \implies c = -3

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Connections

Prerequisite: Completing the Square — the quadratic formula (from which the discriminant arises) is derived directly by completing the square on $ax^2 + bx + c = 0$ . Understanding this derivation gives the discriminant its meaning.

Closely linked within Quadratics:

Solving Quadratic Equations — the discriminant predicts what the solving process will yield.
Quadratic Inequalities — the sign of the discriminant and the factorisation of $\Delta$ as a quadratic in an unknown constant both require inequality-solving techniques.

Later topics that use the discriminant:

Coordinate Geometry: Lines and Curves — tangency conditions between lines and curves reduce to $\Delta = 0$ on the resulting quadratic.
Functions — determining the range of a quadratic function often involves setting $\Delta \geq 0$ and solving for the output variable.
Further Pure topics — in complex numbers (A2), the discriminant underpins why complex roots occur in conjugate pairs for real-coefficient polynomials.

The Discriminant of a Quadratic — CAIE A-Level Pure Mathematics 1 (9709)

Introduction

Core Concept

Key Formulae & Definitions

Worked Examples

Example 1 — Determining the nature of roots

Example 2 — Finding an unknown constant given a repeated root

Example 3 — Using the discriminant to find a range of values

Common Mistakes & Examiner Pitfalls

Practice Questions

Connections

Figures

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