Discriminant and Nature of Roots (Year 11)

01 / LEARN

Understand the idea.

Interpret the square root

The discriminant $Δ = b^{2} - 4 a c$ controls the square root in the quadratic formula. For $a \neq 0$ , positive $Δ$ gives two distinct real roots, zero gives one repeated real root, and negative $Δ$ gives no real roots.

Real roots are horizontal-axis intercepts of the parabola. A repeated root occurs where the vertex touches that axis. With no real roots, the continuous parabola stays entirely above or entirely below it; the graph still exists.

Δ = b^{2} - 4 a c

Check the leading coefficient

For a parameter question, substitute the coefficients into the discriminant and solve the required equality or inequality. Check first whether any parameter value makes $a = 0$ : that value produces a linear or constant equation and needs separate treatment.

The sign of the discriminant counts real roots, but does not make them integers. With rational coefficients, a positive rational square discriminant gives rational roots. Otherwise the real roots can contain surds.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Count without solving

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

Worked solution

1
Identify the quadratic coefficients.
$a = 2, b = 3, c = 5$
All terms are already on one side and the leading coefficient is nonzero.
2
Calculate the discriminant.
$Δ = 3^{2} - 4 (2) (5) = 9 - 40 = - 31$
The signed quantity under the formula’s square root determines whether its roots are real.
3
Classify the roots.
$Δ < 0$
A negative number has no real square root, so the quadratic has no real roots and its parabola has no horizontal-axis intercepts.

Answer

No real roots.

EXAMPLE 02

A repeated-root parameter

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

Worked solution

1
Impose the repeated-root condition.
$Δ = {(- 6)}^{2} - 4 (1) k = 36 - 4 k = 0$
The leading coefficient is always one, so this remains a quadratic for every value of k.
2
Solve for the parameter.
$4 k = 36 ⟹ k = 9$
A repeated real root requires the discriminant to equal zero.
3
Find and confirm the repeated root.
$x^{2} - 6 x + 9 = {(x - 3)}^{2} = 0 ⟹ x = 3$
The square factor occurs twice, giving one distinct root with multiplicity two.

Answer

$k = 9$ ; root x=3.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

One repeated real root.

The discriminant is $4^{2} - 4 (1) (4) = 0$ . Equivalently, the equation is ${(x + 2)}^{2} = 0$ , so there is one repeated real root at $x = - 2$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet 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 $64 - 4 k = 0$ .
Solve for $k$ .
$k = 16$
For which real $k$ does $x^{2} + 2 x + k = 0$ have two distinct real roots?
3 marks
Require $4 - 4 k > 0$ .
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 $x^{2}$ vanishes.
$k = 1$ : repeated root -1; $k = 0$ : linear equation with root $\frac{- 1}{2}$ .

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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 ↗

Discriminant and the nature of roots

In this lesson

Understand the idea.

Interpret the square root

Check the leading coefficient

See the method in action.

Count without solving

Identify the quadratic coefficients.

Calculate the discriminant.

Classify the roots.

A repeated-root parameter

Impose the repeated-root condition.

Solve for the parameter.

Find and confirm the repeated root.

Try it yourself.

Check your understanding

Keep practising