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How does the discriminant tell you the number and nature of the roots of a quadratic equation?

Using the discriminant b squared minus 4ac to determine the number and nature of the roots of a quadratic equation.

A focused answer to the SQA National 5 Mathematics discriminant content, covering how the value of b squared minus 4ac determines whether a quadratic has two real roots, one repeated root or no real roots, and what this means for the graph.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-14

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
What the discriminant is
The three cases
Using the discriminant in problems
Why the discriminant works
Examples in context
Try this

What this dot point is asking

The SQA wants you to calculate the discriminant $b^2 - 4ac$ of a quadratic equation and use its value to state the number and nature of the roots: two distinct real roots, one repeated (equal) root, or no real roots. You should also connect this to how many times the parabola meets the x-axis.

What the discriminant is

The discriminant is the part of the quadratic formula that sits under the square root: $b^2 - 4ac$ . Because you cannot take the real square root of a negative number, the sign of this expression decides how many real solutions the equation has, before you do any further work.

The three cases

The value of the discriminant splits into three cases, each with a clear meaning for the roots and for the graph.

A perfect-square discriminant (such as $9$ or $16$ ) is a special case of the first: there are two distinct real roots, and they are rational, so the quadratic factorises.

Find the nature of the roots of

x^2 + 3x + 1 = 0

The discriminant $b^2 - 4ac$ is the expression under the square root in the quadratic formula. Its sign tells us immediately how many real solutions exist, without our having to complete the formula.

Step 1: Identify the coefficients

Match the equation to $ax^2 + bx + c = 0$ :

a = 1, \quad b = 3, \quad c = 1

Step 2: Calculate the discriminant

Substitute into $b^2 - 4ac$ :

3^2 - 4 \times 1 \times 1 = 9 - 4 = 5

Step 3: Interpret the result

Since $5 > 0$ , the discriminant is positive, so the equation has two distinct real roots. Because $5$ is not a perfect square, $\sqrt{5}$ is irrational, so the roots are irrational.

Final answer: Two distinct real (irrational) roots.

Using the discriminant in problems

Some questions give an unknown coefficient and a condition on the roots, then ask you to find the unknown. You set up the discriminant and apply the condition.

For what value of

k

does

x^2 + kx + 9 = 0

have equal roots?

Equal roots occur precisely when the discriminant equals zero, because a zero discriminant means $\sqrt{b^2 - 4ac} = 0$ and the $\pm$ gives the same value twice. Setting up and solving the resulting equation finds $k$ .

Step 1: Set the discriminant equal to zero

With $a = 1$ , $b = k$ and $c = 9$ , the equal-roots condition is:

b^2 - 4ac = 0 \implies k^2 - 4 \times 1 \times 9 = 0

k^2 - 36 = 0

Step 2: Solve for k

Rearrange and take the square root, remembering both positive and negative solutions:

k^2 = 36 \implies k = 6 \text{ or } k = -6

Final answer: $k = 6$ or $k = -6$ .

Other questions use an inequality on the discriminant. "Two distinct real roots" means $b^2 - 4ac > 0$ , and "no real roots" means $b^2 - 4ac < 0$ , each giving an inequality to solve for the unknown.

Find the values of

p

for which

x^2 + px + 4 = 0

has no real roots

No real roots exist when the discriminant is strictly negative, because a negative value under the square root has no real solution. This gives an inequality in $p$ to solve.

Step 1: Write the no-roots condition as an inequality

With $a = 1$ , $b = p$ and $c = 4$ , the condition $b^2 - 4ac < 0$ becomes:

p^2 - 4 \times 1 \times 4 < 0

p^2 - 16 < 0

Step 2: Solve the inequality

Add $16$ to both sides:

p^2 < 16

Taking square roots of both sides of an inequality $p^2 < 16$ gives a double inequality: $p$ must lie strictly between $-4$ and $4$ :

-4 < p < 4

Final answer: Any value of $p$ with $-4 < p < 4$ gives no real roots.

Why the discriminant works

In the quadratic formula the roots are $\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The two roots differ only in the $\pm$ in front of the square root. If $b^2 - 4ac$ is positive the root is a real number and the $\pm$ gives two different answers; if it is zero the root is zero so both answers coincide; if it is negative there is no real square root, so there are no real answers. The discriminant is simply that decisive quantity under the root.

Examples in context

The discriminant answers "does this ever happen" questions without finding when. If a projectile's height is modelled by a quadratic, a negative discriminant for the equation "height $=$ target" shows the projectile never reaches that target. Engineers test whether a design equation has real solutions before committing to detailed calculation, using the sign of $b^2 - 4ac$ as a fast feasibility check.

Try this

Q1. Find the discriminant of $x^2 + 6x + 5 = 0$ and state the nature of the roots. [2 marks]

Cue. $36 - 20 = 16 > 0$ , two distinct real roots.

Q2. State the nature of the roots of $x^2 - 2x + 5 = 0$ . [2 marks]

Cue. $4 - 20 = -16 < 0$ , no real roots.

Q3. Find $k$ if $x^2 + kx + 16 = 0$ has equal roots. [3 marks]

Cue. $k^2 - 64 = 0$ , so $k = \pm 8$ .

Exam-style practice questions

Practice questions written in the style of SQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

SQA National 5 20193 marksUse the discriminant to determine the nature of the roots of

x^2 - 4x + 4 = 0

Show worked answer →

Identify $a = 1$ , $b = -4$ , $c = 4$ (1 mark). Calculate the discriminant $b^2 - 4ac = (-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0$ (1 mark). Because the discriminant is zero, the equation has one repeated real root (equal roots) (1 mark). Markers reward the discriminant value of $0$ and the conclusion of equal roots.

SQA National 5 20223 marksShow that the equation

2x^2 + x + 3 = 0

has no real roots.

Show worked answer →

Identify $a = 2$ , $b = 1$ , $c = 3$ (1 mark). Calculate $b^2 - 4ac = 1^2 - 4 \times 2 \times 3 = 1 - 24 = -23$ (1 mark). Since the discriminant is negative ( $-23 < 0$ ), the equation has no real roots (1 mark). Markers reward the negative discriminant and the statement that there are no real roots.

Related dot points

Sources & how we know this

SQA National 5 Mathematics Course Specification — SQA (2018)