Nature of Roots

Q: What is the difference between real and non-real roots?

Real roots are real numbers, so the parabola meets the $x$-axis. Non-real roots mean it does not.

The discriminant tells you what kind of roots a quadratic has without solving it. This idea comes back in Grade 12 functions and in tangent questions, so it is worth understanding well.

The discriminant

For $a x^{2} + b x + c = 0$ , the discriminant is

$Δ = b^{2} - 4 a c$

It is the expression under the root in the quadratic formula, so its sign decides whether the formula gives real answers.

Example. $2 x^{2} - 3 x + 5 = 0$ : $Δ = 9 - 40 = - 31$ .

Real, non-real, equal, unequal, rational and irrational roots

$Δ < 0$ : non-real roots
$Δ = 0$ : real, equal (and rational) roots
$Δ > 0$ and a perfect square: real, unequal, rational roots
$Δ > 0$ and not a perfect square: real, unequal, irrational roots

(For equations with rational coefficients.)

Examples.

$x^{2} - 6 x + 9 = 0$ : $Δ = 0$ , equal roots ( $x = 3$ )
$2 x^{2} + 7 x + 3 = 0$ : $Δ = 25$ , real, unequal, rational roots
$x^{2} - 3 x + 1 = 0$ : $Δ = 5$ , real, unequal, irrational roots

On a graph, $Δ$ tells you whether the parabola cuts the $x$ -axis twice, touches it, or misses it.

Finding unknown values for given root types

Write the condition on $Δ$ and solve for the unknown.

Equal roots: $Δ = 0$
Real roots: $Δ \geq 0$
Non-real roots: $Δ < 0$

Example. For which $k$ does $k x^{2} - 4 x + 2 = 0$ have real roots?

$16 - 8 k \geq 0 \Rightarrow k \leq 2$

For the equation to be quadratic, $k \neq = 0$ as well.

Proving roots are always real. Show that $Δ$ is never negative, for example by writing it as a square. If the coefficients are rational and $Δ$ is a perfect square, the roots are rational as well. With a letter in the coefficients, that only holds when the letter stands for a rational number.

Key terms

Discriminant: $Δ = b^{2} - 4 a c$ .
Rational: A number that can be written as a fraction of integers.
Irrational: A real number that cannot be written as a fraction, such as $5$ .
Non-real: Not a real number; occurs when $Δ < 0$ .

Common mistakes

Reading $a$ , $b$ and $c$ before writing the equation in standard form.
Saying 'no roots' instead of 'non-real roots' when $Δ < 0$ .
Using $Δ > 0$ when the question asks for real roots ( $Δ \geq 0$ ).
Forgetting $k \neq = 0$ when $k$ is the coefficient of $x^{2}$ .

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

Determine the nature of the roots of $2 x^{2} - 3 x + 5 = 0$ .

Worked solution

Hidden - try the question first

Solution

$Δ = (- 3)^{2} - 4 (2) (5)$
$= - 31$
$Δ < 0$ : non-real roots

Question 2

5 marks total

routine

Without solving, describe the roots of:

(a) $x^{2} - 6 x + 9 = 0$ (2)

(b) $2 x^{2} + 7 x + 3 = 0$ (3)

Worked solution

Hidden - try the question first

Solution

(a) $Δ = 36 - 36 = 0$
Real, equal and rational
(b) $Δ = 49 - 24$
$= 25$
Positive perfect square: real, unequal and rational

Question 3

6 marks total

complex

For which values of $k$ does $k x^{2} - 4 x + 2 = 0$ have real roots?

Worked solution

Hidden - try the question first

Solution

Real roots: $Δ = 36 - 36 = 0$
$(- 4)^{2} - 4 (k) (2) \geq 0$
$16 - 8 k \geq 0$
$k \leq 2$
For a quadratic, $k \neq = 0$
$k \leq 2$ , $k \neq = 0$

Question 4

8 marks total

problem-solving

Given $x^{2} - 2 m x + m^{2} - 1 = 0$ .

(a) Show that the roots are real and unequal for all real values of $m$ , and rational when $m$ is rational. (4)

(b) Determine the roots in terms of $m$ . (2)

Worked solution

Hidden - try the question first

Solution

(a) $Δ = 36 - 36 = 0$
$= 4 m^{2} - 4 m^{2} + 4$
$= 4$
$Δ = 4 > 0$ , so the roots are real and unequal; $Δ = 2$ , so for rational $m$ the roots are rational
(b) $x = \frac{2 m \pm 2}{2}$
$x = m + 1$ or $x = m - 1$
(c) The smaller root must be positive: $m - 1 > 0$
$m > 1$

FAQs

What does the discriminant tell me?

Whether the roots are real or non-real, equal or unequal, and rational or irrational.

What is the difference between real and non-real roots?

Real roots are real numbers, so the parabola meets the $x$ -axis. Non-real roots mean it does not.

Is nature of roots tested in Grade 12?

Yes, especially inside function and tangent questions.

The discriminant

Real, non-real, equal, unequal, rational and irrational roots

Finding unknown values for given root types

Key terms

Common mistakes

Practice questions

FAQs

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