Discriminant and the nature of roots | Year 11

01 / LEARN

The idea, step by step

Interpret the square root

The expression under the square root determines real root behaviour. Positive gives two distinct roots, zero one repeated root, and negative no real roots.

Roots are x-intercepts. A repeated root means the parabola touches the x-axis at its vertex. With no real roots, it stays entirely above or entirely below the axis.

Δ = b^{2} - 4 a c

Check the leading coefficient

For parameter problems, form the discriminant and solve its equality or inequality. Check separately that the leading coefficient is not zero. Otherwise the equation may have become linear.

A positive perfect-square discriminant with rational coefficients gives rational roots. The sign alone determines the count, not whether those roots are integers.

A common trap

Saying a negative discriminant means there is no graph.

Instead: The graph exists but has no real x-intercepts.

02 / WORKED EXAMPLES

See the thinking

Example 1

Count without solving

How many real roots has $2 x^{2} + 3 x + 5 = 0$ ?

$Δ = 9 - 4 (2) (5) = - 31$ .
The discriminant is negative.

No real roots.

Example 2

A repeated-root parameter

Find k for a repeated root in $x^{2} - 6 x + k = 0$ .

Require $Δ = 36 - 4 k = 0$ .
Solve k=9.
Check ${(x - 3)}^{2} = 0$ .

k = 9

; root x=3.

Check your understanding

Classify the roots of $x^{2} + 4 x + 4 = 0$ .

Show answer

One repeated real root.

The discriminant is zero.

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

How many real roots has $x^{2} - 7 x + 10 = 0$ ?
2 marks
How many real roots has $3 x^{2} + 2 x + 4 = 0$ ?
2 marks
Find k for a repeated root in $x^{2} + 8 x + k = 0$ .
2 marks
For which real k does $x^{2} + 2 x + k = 0$ have two distinct real roots?
3 marks
Find k for a repeated root in $x^{2} + k x + 9 = 0$ .
3 marks
Classify $k x^{2} + 2 x + 1 = 0$ at k=1 and at k=0.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

How many real roots has $x^{2} - 7 x + 10 = 0$ ?
2 marks
The discriminant is 49-40=9, positive.
Two distinct real roots.
How many real roots has $3 x^{2} + 2 x + 4 = 0$ ?
2 marks
The discriminant is 4-48=-44.
No real roots.
Find k for a repeated root in $x^{2} + 8 x + k = 0$ .
2 marks
Set 64-4k=0.
Solve for k.
$k = 16$
For which real k does $x^{2} + 2 x + k = 0$ have two distinct real roots?
3 marks
Require 4-4k>0.
Rearrange the inequality.
$k < 1$
Find k for a repeated root in $x^{2} + k x + 9 = 0$ .
3 marks
Require $k^{2} - 36 = 0$ .
Keep both square-root branches.
$k = - 6$ or 6.
Classify $k x^{2} + 2 x + 1 = 0$ at k=1 and at k=0.
3 marks
At 1, the discriminant is zero.
At 0, the coefficient of x squared vanishes.
k=1: repeated root -1; k=0: linear equation with root -1/2.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics topics 2.1–2.2: quadratic relationships, cubic and quartic polynomials. Original Ascent teaching sequence.

Official curriculum source ↗

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