MathsISC Class 11 · Quadratic Equations

Quadratic Equations: Discriminant, Nature of Roots, Sum and Product of Roots

Q: Why are complex roots always in conjugate pairs?

For real coefficients, the quadratic formula gives p + iq and p − iq.

Q: What is a double root?

Two equal roots: the graph touches the x-axis at one point.

Q: How do I know if the roots are rational?

If a, b, c are rational and D is a perfect square.

Updated 17 October 2026 · Fact-checked

For ax² + bx + c = 0 the roots are x = (−b ± √D)/2a with D = b² − 4ac. If D > 0 the roots are real and distinct, D = 0 gives equal real roots, and D < 0 gives a pair of complex conjugate roots. The sum of the roots is −b/a and the product is c/a, from which symmetric expressions and new equations are built.

The sign of D = b² − 4ac tells how many times the parabola meets the x-axis: twice, once (touching), or never (complex roots).

Illustration drawn for QuestionBuddy.

Understand Nature of Roots and Relations between the Roots

Formula. The roots of $a x^{2} + b x + c = 0$ (a ≠ 0) are $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ . The expression $D = b^{2} - 4 a c$ is the discriminant.

Nature. $D > 0$ : two distinct real roots (rational if D is a perfect square and a, b, c are rational). $D = 0$ : equal real roots $x = - \frac{b}{2 a}$ . $D < 0$ : no real roots; the roots are complex conjugates $p \pm i q$ . If a, b, c are rational and D > 0 is not a perfect square, the roots are irrational conjugates.

Relations between the roots α and β. $α + β = - \frac{b}{a}$ and $α β = \frac{c}{a}$ . The equation with roots α and β is $x^{2} - (α + β) x + α β = 0$ . Special conditions: roots equal in magnitude and opposite in sign: $b = 0$ ; roots reciprocal: $c = a$ ; one root zero: $c = 0$ .

Symmetric expressions. $α^{2} + β^{2} = (α + β)^{2} - 2 α β$ ; $α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)$ ; $(α - β)^{2} = (α + β)^{2} - 4 α β$ ; $\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}$ .

Forming a new equation. To find the equation whose roots are functions of α, β (for example α + 2, β + 2, or α², β²), compute the new sum S and new product P using the sum and product of α, β, and write $x^{2} - S x + P = 0$ .

Equal roots. For a real parameter k, set $D = 0$ and solve for k.

Nature of Roots and Relations between the Roots: formulas to remember

Roots: $x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$; D = b² − 4ac.
Nature of roots: $D > 0 r e a l, d i s t in c t ∣ D = 0 r e a l, e q u a l ∣ D < 0 co m pl e x co nj ug a t es$; Rational roots need D a perfect square.
Sum and product: $α + β = -\frac{b}{a}$ | $αβ = \frac{c}{a}$; Equation: $x^{2} - (α + β) x + α β = 0$ .
Symmetric forms: $$α^2 + β^2 = (α + β)^2 - 2αβ$ | $α^3 + β^3 = (α + β)^3 - 3αβ(α + β)$$; $∣ α - β ∣ = \frac{D}{∣ a ∣}$ .

Nature of Roots and Relations between the Roots in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Time of flight of a ball

Suppose the height of a ball thrown up is $h = 20 t - 5 t^{2}$ metres, and we want the times at which it is at 15 m.

Solve $20 t - 5 t^{2} = 15$ and comment on the roots.

Try it first, then show the solutionSolution

$5 t^{2} - 20 t + 15 = 0$ , i.e. $t^{2} - 4 t + 3 = 0$ .
$D = 16 - 12 = 4 > 0$ : roots $t = \frac{4 \pm 2}{2} = 1$ or $3$ .
The ball is at 15 m at t = 1 s (going up) and t = 3 s (coming down).

Answer: t = 1 s and t = 3 s.

Memory hook: Two real roots mean two times at that height.

Tricks to remember Nature of Roots and Relations between the Roots

Short memory aids. Say each one aloud once, then use it on a question the same day.

D tells the story
Compute D before solving, to know the nature of the roots.
Use it when: Nature-of-roots questions.
Sum and product first
Never find the actual roots if only symmetric expressions are needed.
Use it when: Saving time.

Nature of Roots and Relations between the Roots: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Discriminant · 1 mark · MCQBoard level

The discriminant of $2 x^{2} - 4 x + 3 = 0$ is:
(a) 8
(b) −8
(c) 40
(d) −40

Try it first, then show the solutionSolution

$D = 16 - 24 = - 8$ , so the roots are complex.

Answer: (b) −8

Example 2: Solving and nature of roots · 3 marks · Short answerBoard level

Solve $x^{2} - 5 x + 6 = 0$ and determine the nature of the roots of $2 x^{2} - 3 x + 5 = 0$ .

Try it first, then show the solutionSolution

$(x - 2) (x - 3) = 0$ gives $x = 2$ or $3$ . (1 mark)
For $2 x^{2} - 3 x + 5$ : $D = 9 - 40 = - 31 < 0$ . (1 mark)
The roots are complex conjugates, $x = \frac{3 \pm i 31}{4}$ . (1 mark)

Answer: 2 and 3; complex conjugate roots.

Example 3: Equal roots · 3 marks · Short answerHarder board level

Find k so that $x^{2} + k x + 9 = 0$ has equal roots, and the roots in each case. For which k are the roots real?

Try it first, then show the solutionSolution

Equal roots: $D = k^{2} - 36 = 0$ , so $k = \pm 6$ . (1 mark)
For $k = 6$ the root is $- 3$ (repeated); for $k = - 6$ the root is $3$ . (1 mark)
Real roots: $k^{2} \geq 36$ , so $k \leq - 6$ or $k \geq 6$ . (1 mark)

Answer: k = ±6; roots ∓3; real for |k| ≥ 6.

Example 4: Symmetric functions · 4 marks · Short answerHarder board level

If α and β are the roots of $2 x^{2} - 5 x + 3 = 0$ , find $α^{2} + β^{2}$ , $α^{3} + β^{3}$ and $∣ α - β ∣$ . Verify with the actual roots.

Try it first, then show the solutionSolution

$α + β = \frac{5}{2}$ and $α β = \frac{3}{2}$ . (0.5 mark)
$α^{2} + β^{2} = \frac{25}{4} - 3 = \frac{13}{4}$ . (1 mark)
$α^{3} + β^{3} = (\frac{5}{2})^{3} - 3 \cdot \frac{3}{2} \cdot \frac{5}{2} = \frac{125}{8} - \frac{45}{4} = \frac{35}{8}$ . (1.5 marks)
$∣ α - β ∣ = \frac{25}{4} - 6 = \frac{1}{2}$ ; the roots are 1 and 3/2. (1 mark)

Answer: 13/4; 35/8; 1/2.

Example 5: Forming new equations · 5 marks · Long answerHOTS / competency

If α, β are the roots of $x^{2} - 4 x + 1 = 0$ , form the equations whose roots are (i) α + 2 and β + 2, (ii) α² and β², (iii) 1/α and 1/β.

Try it first, then show the solutionSolution

$α + β = 4$ and $α β = 1$ . (0.5 mark)
(i) Sum $= 8$ , product $= α β + 2 (α + β) + 4 = 13$ : $x^{2} - 8 x + 13 = 0$ . (1.5 marks)
(ii) Sum $= 16 - 2 = 14$ , product $= 1$ : $x^{2} - 14 x + 1 = 0$ . (1.5 marks)
(iii) Sum $= \frac{α + β}{α β} = 4$ , product $= 1$ : $x^{2} - 4 x + 1 = 0$ (the same equation). (1.5 marks)

Answer: x² − 8x + 13 = 0; x² − 14x + 1 = 0; x² − 4x + 1 = 0.

Where students lose marks in Nature of Roots and Relations between the Roots

The slips examiners see most often, and what to write instead.

Mistake: Wrong sign in −b/a
Taking the sum of roots as b/a.
Do this: The sum is −b/a.
Usually costs: 2 marks.
Mistake: Forgetting that D < 0 gives complex roots
Saying "no roots".
Do this: There are no real roots but two complex conjugate roots.
Usually costs: 1 mark.
Mistake: Expanding symmetric expressions incorrectly
Writing α² + β² = (α + β)².
Do this: α² + β² = (α + β)² − 2αβ.
Usually costs: 2 marks.

Exam tips

Write the equation in the form ax² + bx + c = 0 first.
Compute D before the roots.
Use sum and product for symmetric expressions.
Verify with one actual root.

Practice questions on Nature of Roots and Relations between the Roots

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
If the roots of $x^{2} + p x + q = 0$ are equal, then:
(a) $p^{2} = 4 q$
(b) $p = q$
(c) $q = 0$
(d) $p^{2} = q$
Hint
D = 0.
Answer
(a) $p^{2} = 4 q$
Q2 · 2 marks · Short answerBoard level
Find the sum and product of the roots of $3 x^{2} - 7 x + 2 = 0$ .
Hint
−b/a and c/a.
Answer
Sum $= \frac{7}{3}$ ; product $= \frac{2}{3}$ .
Q3 · 3 marks · Short answerHarder board level
Find the nature of roots of $x^{2} - 6 x + 7 = 0$ and solve it.
Hint
D = 36 − 28.
Answer
$D = 8 > 0$ , not a perfect square: irrational roots $x = 3 \pm 2$ .
Q4 · 3 marks · Short answerHarder board level
If one root of $x^{2} - 7 x + k = 0$ is twice the other, find k.
Hint
Let the roots be α and 2α.
Answer
$3 α = 7$ gives $α = \frac{7}{3}$ ; product $2 α^{2} = k$ , so $k = 2 \times \frac{49}{9} = \frac{98}{9}$ .
Q5 · 3 marks · Short answerHarder board level
Form a quadratic equation with roots $3 + i$ and $3 - i$ .
Hint
Sum and product.
Answer
Sum $= 6$ ; product $= 9 + 1 = 10$ : $x^{2} - 6 x + 10 = 0$ .
Q6 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a rectangular garden has a perimeter of 28 m and an area of 48 m².
(i) Write the quadratic for the length. (1 mark)
(ii) Find its discriminant. (1 mark)
(iii) Find the dimensions. (1 mark)
(iv) Can the area be 60 m² with the same perimeter? (1 mark)
Hint
Length + breadth = 14.
Answer
(i) $l (14 - l) = 48$ , i.e. $l^{2} - 14 l + 48 = 0$ . (ii) $D = 196 - 192 = 4$ . (iii) $l = 8$ , breadth 6 (or the reverse). (iv) $l^{2} - 14 l + 60 = 0$ has $D = 196 - 240 < 0$ : no real solution, so 60 m² is not possible.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.D?Show answer
b² − 4ac.
2.Sum of roots?Show answer
−b/a.
3.Product of roots?Show answer
c/a.
4.Equal roots condition?Show answer
D = 0.

Nature of Roots and Relations between the Roots: frequently asked questions

Why are complex roots always in conjugate pairs?

For real coefficients, the quadratic formula gives p + iq and p − iq.

What is a double root?

Two equal roots: the graph touches the x-axis at one point.

How do I know if the roots are rational?

If a, b, c are rational and D is a perfect square.