Nature of Roots and the Discriminant | IEB Grade 12 Maths

Q: What does the discriminant tell me?

Whether the roots are real or non-real, equal or unequal, and rational or irrational, without solving the equation.

Q: Where else is the nature of roots tested?

In functions and analytical geometry, to decide whether a line cuts, touches or misses a curve, and in tangent questions.

Q: Do real roots include equal roots?

Yes. 'Real roots' means $\Delta \ge 0$, which includes the equal-roots case.

The discriminant tells you what kind of roots a quadratic has without solving it. In Grade 12 it is tested on its own and inside function and tangent questions, often to find the value of an unknown constant.

Using the discriminant to describe roots

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

$Δ = b^{2} - 4 a c$

$Δ < 0$ : 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

This assumes $a$ , $b$ and $c$ are rational.

Example. $3 x^{2} - 5 x + 1 = 0$ : $Δ = 25 - 12 = 13$ . This is positive and not a perfect square, so the roots are real, unequal and irrational.

On a graph. $Δ$ tells you how many times the parabola $y = a x^{2} + b x + c$ cuts the $x$ -axis: twice, once (touches) or not at all.

Finding parameter values for given root types

$Δ = (p - 2)^{2} \geq 0$ , so the roots are real for all real $p$ ; for rational $p$ , $Δ = ∣ p - 2∣$ is rational, so the roots are rational

Key terms

Discriminant: $Δ = b^{2} - 4 a c$ , the expression under the root in the quadratic formula.
Rational roots: Roots that can be written as fractions of integers.
Irrational roots: Real roots that cannot be written as fractions, such as $13$ .
Non-real roots: Roots that are not real numbers, when $Δ < 0$ .

Common mistakes

Not writing the equation in standard form before reading $a$ , $b$ and $c$ .
Saying $Δ = 0$ gives 'no roots' instead of equal roots.
Forgetting both signs when solving $k^{2} = 64$ .
Using $Δ > 0$ when the question asks for real roots, which includes $Δ = 0$ .

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 $3 x^{2} - 5 x + 1 = 0$ without solving the equation.

Worked solution

Hidden - try the question first

Solution

$Δ = (- 5)^{2} - 4 (3) (1)$
$= 13$
$Δ > 0$ and not a perfect square: real, unequal and irrational

Question 2

5 marks total

routine

For which values of $k$ does $x^{2} + k x + 16 = 0$ have equal roots?

Worked solution

Hidden - try the question first

Solution

Equal roots: $Δ = 0$
$k^{2} - 4 (1) (16) = 0$
$k^{2} = 64$
$k = 8$
or $k = - 8$

Question 3

6 marks total

complex

Prove that the roots of $x^{2} - (p + 2) x + 2 p = 0$ are real for all real values of $p$ , and rational for all rational values of $p$ .

Worked solution

Hidden - try the question first

Solution

$Δ = (- 5)^{2} - 4 (3) (1)$
$Δ > 0$
$= p^{2} - 4 p + 4$
$= (p - 2)^{2}$
$(p - 2)^{2} \geq 0$ for all real $p$ , so the roots are real
For rational $p$ the coefficients are rational and $Δ = ∣ p - 2∣$ is rational, so the roots are rational

Question 4

9 marks total

problem-solving

The line $y = m x - 1$ and the parabola $y = x^{2} + 3$ are given.

(a) For which values of $m$ will the line not meet the parabola? (5)

(b) For which values of $m$ is the line a tangent to the parabola? Determine the point of contact in each case. (4)

Worked solution

Hidden - try the question first

Solution

(a) $Δ = 0$
No intersection: $Δ < 0$
$m^{2} - 16 < 0$
$(m - 4) (m + 4) < 0$
$- 4 < m < 4$
(b) Tangent: $Δ = 0$ , so $m = \pm 4$
$m = 4$ : $x^{2} - 4 x + 4 = 0 \Rightarrow x = 2$ , $y = 7$
$m = - 4$ : $x^{2} + 4 x + 4 = 0 \Rightarrow x = - 2$ , $y = 7$
Points of contact $(2; 7)$ and $(- 2; 7)$

FAQs

What does the discriminant tell me?

Whether the roots are real or non-real, equal or unequal, and rational or irrational, without solving the equation.

Where else is the nature of roots tested?

In functions and analytical geometry, to decide whether a line cuts, touches or misses a curve, and in tangent questions.

Do real roots include equal roots?

Yes. 'Real roots' means $Δ \geq 0$ , which includes the equal-roots case.

Nature of roots

Using the discriminant to describe roots

Finding parameter values for given root types

Key terms

Common mistakes

Practice questions

FAQs

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