Discriminant and number of real roots | Year 10

01 / LEARN

Understand the idea.

The square root decides the outcome

The discriminant, $Δ = b^{2} - 4 a c$ , determines whether the square root in the quadratic formula is positive, zero or unavailable over the real numbers. This classifies the roots without calculating them.

For $a \neq 0$ , a positive discriminant gives two distinct real roots; zero gives one repeated real root; a negative discriminant gives none. The graph respectively crosses twice, touches once, or misses the horizontal axis.

\begin{array}{c} Δ = b^{2} - 4 a c \\ \begin{aligned} Δ > 0 & : two real roots \\ Δ = 0 & : one repeated real root \\ Δ < 0 & : no real roots \end{aligned} \end{array}

Keep the question and domain in view

When a coefficient is unknown, a repeated-root condition becomes the equation $Δ = 0$ . Solving it identifies where two real roots merge into a single point of contact with the horizontal axis.

Root counts describe the equation over all real inputs. They do not decide how many solutions fit a model: a negative time, an impossible length or a value outside the stated interval must still be excluded.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Classify without solving

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

Worked solution

1
Identify the signed coefficients.
$a = 3, b = - 5, c = - 2$
The equation is in standard quadratic form with a nonzero leading coefficient.
2
Calculate the discriminant.
$\begin{aligned} Δ & = {(- 5)}^{2} - 4 (3) (- 2) \\ = 25 + 24 = 49 \end{aligned}$
The negative constant makes the subtracted product $4 a c$ negative.
3
Classify using its sign.
$Δ = 49 > 0$
A positive square root gives two different branches in the quadratic formula, so there are two distinct real roots.

Answer

Two distinct real roots.

EXAMPLE 02

Find the boundary value

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

Worked solution

1
Express the discriminant in terms of the parameter.
$Δ = {(- 6)}^{2} - 4 (1) k = 36 - 4 k$
The constant coefficient is the unknown $k$ .
2
Impose the repeated-root condition.
$\begin{aligned} 36 - 4 k & = 0 \\ 4 k & = 36 \\ k & = 9 \end{aligned}$
A repeated real root occurs exactly when the discriminant is zero.
3
Factor the resulting quadratic.
$x^{2} - 6 x + 9 = {(x - 3)}^{2}$
The two identical linear factors show why the root is repeated.
4
State the parameter and root.
$k = 9, {(x - 3)}^{2} = 0 ⟹ x = 3$
The parameter value and the repeated solution answer different parts of the question.

Answer

$k = 9$ , with repeated root $x = 3$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

No real roots.

Here $a = 1$ , $b = 2$ and $c = 5$ , so $Δ = 2^{2} - 4 (1) (5) = - 16 < 0$ . The equation therefore has no real roots.

Keep practising

6 worksheet questions · 16 marks

Open worksheet 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.

Official curriculum source ↗

In this lesson

Understand the idea.

The square root decides the outcome

Keep the question and domain in view

See the method in action.

Classify without solving

Identify the signed coefficients.

Calculate the discriminant.

Classify using its sign.

Find the boundary value

Express the discriminant in terms of the parameter.

Impose the repeated-root condition.

Factor the resulting quadratic.

State the parameter and root.

Try it yourself.

Check your understanding

Keep practising