Discriminant and number of real roots | Year 10

01 / LEARN

The idea, step by step

The square root decides the outcome

In the quadratic formula, the expression under the square root is called the discriminant. A positive value gives two different real square-root choices. Zero makes the two choices equal; a negative value gives no real roots.

For a graph y=ax squared plus bx plus c, these cases mean two x-axis crossings, one tangent contact, or no x-axis crossing.

Δ = b^{2} - 4 a c : Δ > 0 two real roots; Δ = 0 one repeated real root; Δ < 0 no real roots

Keep the question and domain in view

A repeated root can locate a boundary between crossing and not crossing. Setting the discriminant to zero is therefore a useful way to find an unknown coefficient.

A positive discriminant does not guarantee two useful answers in a real situation. A time may be negative or a length may be outside a stated range; check those conditions separately.

A common trap

Calling a negative discriminant a negative root.

Instead: A negative discriminant means there is no real root, not that the roots themselves are negative.

02 / WORKED EXAMPLES

See the thinking

Example 1

Classify without solving

How many real roots does $3 x^{2} - 5 x - 2 = 0$ have?

$Δ = {(- 5)}^{2} - 4 (3) (- 2) = 25 + 24 = 49$ .
$49 > 0$ , so the two roots are real and different.

Two distinct real roots.

Example 2

Find the boundary value

For what value of k does $x^{2} - 6 x + k = 0$ have one repeated root?

$Δ = 36 - 4 k$ .
A repeated root requires $36 - 4 k = 0$ , so $k = 9$ .
Then $x^{2} - 6 x + 9 = {(x - 3)}^{2}$ .

k = 9

, with repeated root

x = 3

Check your understanding

Classify $x^{2} + 2 x + 5 = 0$ using its discriminant.

Show answer

No real roots.

$Δ = 2^{2} - 4 (1) (5) = - 16 < 0$ .

03 / WORKSHEET

Your turn

6 questions · 16 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the discriminant of $x^{2} - 8 x + 7 = 0$ and state the number of real roots.
2 marks
Classify the real roots of $2 x^{2} - 4 x + 2 = 0$ .
2 marks
Classify the real roots of $2 x^{2} + x + 3 = 0$ .
2 marks
Find c if $x^{2} - 4 x + c = 0$ has one repeated real root.
3 marks
For which real values of k does $x^{2} + k x + 9 = 0$ have two distinct real roots?
3 marks
A graph has equation $y = x^{2} - 6 x + k$ . Describe its x-axis intersections when k is less than, equal to, or greater than 9.
4 marks

Read the worked solutions

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

Find the discriminant of $x^{2} - 8 x + 7 = 0$ and state the number of real roots.
2 marks
${(- 8)}^{2} - 4 (1) (7) = 64 - 28 = 36$ .
$Δ = 36$ ; two distinct real roots.
Classify the real roots of $2 x^{2} - 4 x + 2 = 0$ .
2 marks
$Δ = 16 - 16 = 0$ .
One repeated real root.
Classify the real roots of $2 x^{2} + x + 3 = 0$ .
2 marks
$Δ = 1 - 24 = - 23$ .
No real roots.
Find c if $x^{2} - 4 x + c = 0$ has one repeated real root.
3 marks
$16 - 4 c = 0$ , hence $c = 4$ .
$c = 4$
For which real values of k does $x^{2} + k x + 9 = 0$ have two distinct real roots?
3 marks
$Δ = k^{2} - 36 > 0$ .
This requires $∣ k ∣ > 6$ .
$k < - 6$ or $k > 6$
A graph has equation $y = x^{2} - 6 x + k$ . Describe its x-axis intersections when k is less than, equal to, or greater than 9.
4 marks
The discriminant is $36 - 4 k$ .
Its sign changes at $k = 9$ .
$k < 9$ : two crossings; $k = 9$ : tangent at $(3, 0)$ ; $k > 9$ : no intersection.

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Curriculum reference & lesson scope

Australian Curriculum v9.0 and South Australian Curriculum

Year 10 quadratics, including the fuller school-course algebra requested for Ascent: non-monic factorisation, completing the square and the quadratic formula. These lessons consolidate Year 9 monic quadratics and extend the available solving methods.

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