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CBSE · Class 10 · Mathematics

Quadratic Equations 4.4: Nature of Roots

5 concepts · 40 practice questions with answers · Free to read

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Calculating the discriminant

From daily life

A doctor measures one number to judge health. For a quadratic, one number, the discriminant, tells how its roots behave.

Simple definition

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. Compute it using the signs of a, b and c.

Formula: D = b² − 4ac

reada, b, cb²− 4acD

In the figure: x² − 5x + 6 = 0: a = 1, b = −5, c = 6, so D = 25 − 24 = 1.

Rule 1: Compute D — Square b first, then subtract 4ac.

  • Easy: 2x² − 4x + 3 = 0: D = 16 − 24 = −8.
  • Medium: x² − 6x + 9 = 0: D = 36 − 36 = 0.
  • Exam-type: 3x² + 2x − 1 = 0: D = 4 + 12 = 16.

Rule 2: Standard form first — Rearrange so that the right side is 0 before reading a, b, c.

  • Easy: x² = 4x − 4 ⇒ x² − 4x + 4 = 0: D = 16 − 16 = 0.
  • Medium: 2x² = 5x + 3 ⇒ 2x² − 5x − 3 = 0: D = 25 + 24 = 49.
  • Exam-type: x² − 9 = 0: b = 0, D = 0 + 36 = 36.
Remember

When a and c have opposite signs, −4ac is positive, so D is positive.

Common mistake

Using b² = −25 for b = −5; the square is always positive.

Memory clue

"b squared minus four a c."

Practice questions

Easy
  1. 1The discriminant of −5x² − 5x − 4 = 0 is
  2. 2The discriminant of −4x² + 7x + 5 = 0 is
  3. 3The discriminant of 3x² + 9x − 7 = 0 is
Medium
  1. 4The discriminant of 2x² − 7x − 3 = 0 is
  2. 5The discriminant of −3x² + 5x + 3 = 0 is
  3. 6The discriminant of −x² + 5x + 9 = 0 is
Challenging
  1. 7The discriminant of −x² + 3x + 3 = 0 is
  2. 8The discriminant of 4x² − 9x − 4 = 0 is
Show answers (try them yourself first)
  1. 1−55
  2. 2129
  3. 3165
  4. 473
  5. 561
  6. 661
  7. 721
  8. 8145

Nature of the roots from the discriminant

From daily life

The sign of D works like a traffic signal: positive means two roots, zero means one repeated root, negative means no real root.

Simple definition

D > 0: two distinct real roots. D = 0: two equal real roots. D < 0: no real roots.

Formula: D > 0: 2 roots; D = 0: 1 root; D < 0: none

D > 0two distinctD = 0two equalD < 0no real

In the figure: x² − 5x + 6 has D = 1 > 0 (roots 2, 3); x² − 6x + 9 has D = 0 (root 3 twice); 2x² − 4x + 3 has D = −8 < 0 (no real root).

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Nature of the roots from the discriminant

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Rule 1: Decide the nature — Compute D, then use its sign.

  • Easy: x² + x + 1 = 0: D = 1 − 4 = −3 < 0, no real roots.
  • Medium: x² − 4x + 4 = 0: D = 0, equal roots.
  • Exam-type: 2x² − 7x + 3 = 0: D = 49 − 24 = 25 > 0, distinct roots.

Rule 2: Rational or not — If D is a perfect square, the roots are rational.

  • Easy: D = 25 (perfect square): roots rational.
  • Medium: D = 8: roots involve √8, irrational.
  • Exam-type: D = 0: the single root is rational: −b/2a.
Remember

Always give the conclusion in words, not just the sign of D.

Common mistake

Saying D < 0 means "no roots"; it means no REAL roots.

Memory clue

"Plus two, zero one, minus none."

Practice questions

Easy
  1. 1What is the nature of the roots of x² + 5x + 9 = 0?
  2. 2What is the nature of the roots of x² − 6x + 9 = 0?
  3. 3What is the nature of the roots of 2x² + 4x + 2 = 0?
Medium
  1. 4What is the nature of the roots of −3x² − 6x − 3 = 0?
  2. 5What is the nature of the roots of −3x² + 2x − 2 = 0?
  3. 6What is the nature of the roots of −3x² − 8x − 2 = 0?
Challenging
  1. 7How many distinct real values of x satisfy 2x² + 9x − 7 = 0?
  2. 8How many distinct real values of x satisfy −2x² − 6x − 9 = 0?
Show answers (try them yourself first)
  1. 1No real roots
  2. 2Two equal real roots
  3. 3Two equal real roots
  4. 4Two equal real roots
  5. 5No real roots
  6. 6Two distinct real roots
  7. 72
  8. 80

Finding k for a required nature of roots

From daily life

If a condition on the roots is given, turn it into a condition on D, then solve for k.

Simple definition

Write D in terms of k. Equal roots: D = 0. Real roots: D ≥ 0. Distinct real roots: D > 0. No real roots: D < 0. Solve the equation or inequality.

write Din ksetD = 0 or signsolvefor k

In the figure: x² + kx + 9 = 0 has equal roots when k² − 36 = 0, so k = ±6.

Rule 1: Equal roots (D = 0) — Equate D to 0 and solve.

  • Easy: x² + kx + 9 = 0: k² − 36 = 0, k = ±6.
  • Medium: 2x² − kx + 8 = 0: k² − 64 = 0, k = ±8.
  • Exam-type: kx² − 2kx + 6 = 0: 4k² − 24k = 0, k = 6 (k ≠ 0).

Rule 2: Real or no real roots — Solve the inequality for k.

  • Easy: x² + 4x + k = 0 has real roots: 16 − 4k ≥ 0, k ≤ 4.
  • Medium: x² − 2x + k = 0 has no real roots: 4 − 4k < 0, k > 1.
  • Exam-type: x² + 6x + k = 0 has distinct roots: 36 − 4k > 0, k < 9.
Remember

Remember to exclude k that makes a = 0, and flip the inequality when dividing by a negative.

Common mistake

Not flipping the inequality sign when dividing by −4.

Memory clue

"Write D in k, set the condition, solve."

Practice questions

Easy
  1. 1Find the value of k for which kx² + 4x + 1 = 0 has two equal roots.
  2. 2For what value of k does x² − 10x + k = 0 have equal roots?
  3. 3Find the value of k for which kx² − 4x + 2 = 0 has two equal roots.
Medium
  1. 4For what value of k does x² + 2x + k = 0 have equal roots?
  2. 5For what value of k does x² + 4x + k = 0 have equal roots?
  3. 6For what value of k does x² − 8x + k = 0 have equal roots?
Challenging
  1. 7For which values of k does x² + 2x + k = 0 have two distinct real roots?
  2. 8For which values of k does x² + 6x + k = 0 have real roots?
Show answers (try them yourself first)
  1. 14
  2. 225
  3. 32
  4. 41
  5. 54
  6. 616
  7. 7k < 1
  8. 8k ≤ 9

Finding roots by the quadratic formula

From daily life

When factorising is hard, one formula gives both roots every time.

Simple definition

The quadratic formula: x = (−b ± √D) / 2a with D = b² − 4ac. If D = 0 the two roots are the same, x = −b/2a. If D < 0 there is no real root.

Formula: x = (−b ± √(b² − 4ac)) / 2a

a, b, cD = b²−4acx = (−b ± √D)/2a

In the figure: x² − 5x + 6 = 0: D = 1, x = (5 ± 1)/2 = 3 or 2.

Rule 1: Apply the formula — Put a, b, c with signs, then simplify the root.

  • Easy: 2x² − 7x + 3 = 0: D = 25, x = (7 ± 5)/4 = 3 or 1/2.
  • Medium: x² + 2x − 1 = 0: D = 8, x = (−2 ± √8)/2 = −1 ± √2.
  • Exam-type: x² − 6x + 9 = 0: D = 0, x = 6/2 = 3 (twice).

Rule 2: Check the roots — Use sum −b/a and product c/a.

  • Easy: 2x² − 7x + 3: roots 3, 1/2: sum 7/2 ✓, product 3/2 ✓.
  • Medium: x² + 2x − 1: sum −2 ✓, product (−1+√2)(−1−√2) = 1 − 2 = −1 ✓.
  • Exam-type: Put one root back in the equation.
Remember

The formula needs the equation in standard form with the right side 0.

Common mistake

Dividing only part of the numerator by 2a, e.g. −b ± √D / 2a.

Memory clue

"Minus b plus-minus root D, over two a."

Practice questions

Easy
  1. 1The quadratic formula for ax² + bx + c = 0 is x =
  2. 2Using the quadratic formula, find the larger root of 12x² − 26x − 16 = 0.
  3. 3What is the sum of the roots of x² + 8x + 1 = 0?
Medium
  1. 4What is the sum of the roots of 3x² + 3x − 8 = 0?
  2. 5What is the product of the roots of 2x² + 3x − 9 = 0?
  3. 6Using the quadratic formula, find the smaller root of 4x² + 2x − 6 = 0.
Challenging
  1. 7What is the sum of the roots of 6x² − 4x − 1 = 0?
  2. 8What is the product of the roots of 5x² − 5x + 1 = 0?
Show answers (try them yourself first)
  1. 1(−b ± √(b² − 4ac)) / 2a
  2. 28/3
  3. 3−8
  4. 4−1
  5. 5−9/2
  6. 6−3/2
  7. 72/3
  8. 81/5

Existence problems

From daily life

Can a park of a given perimeter and area exist? Form the equation and let D answer yes or no.

Simple definition

To decide whether a situation is possible, form the quadratic, find D. If D ≥ 0 real values exist; if D < 0 it is not possible. Also check the roots make sense (positive lengths).

formquadraticfindDD ≥ 0possibleD < 0not possible

In the figure: A park with perimeter 80 m and area 400 m²: x² − 40x + 400 = 0, D = 0, so it is possible with length = breadth = 20 m.

Rule 1: Possible or not? — Use the sign of D to answer.

  • Easy: Perimeter 80 m, area 800 m²: x² − 40x + 800 = 0, D = 1600 − 3200 < 0: not possible.
  • Medium: Two numbers with sum 10 and product 30: x² − 10x + 30 = 0, D = 100 − 120 < 0: not possible.
  • Exam-type: Sum 10 and product 21: D = 100 − 84 = 16 > 0, numbers 3 and 7.

Rule 2: Find the values — If possible, solve and keep only sensible roots.

  • Easy: (x − 20)² = 0: length 20 m, breadth 20 m (a square).
  • Medium: Sum 10, product 21: (x − 3)(x − 7) = 0.
  • Exam-type: Always state the conclusion in a sentence.
Remember

D = 0 means exactly one solution (equal roots), often a square or a pair of equal numbers.

Common mistake

Saying "not possible" when D = 0; D = 0 still gives a valid answer.

Memory clue

"D non-negative: it exists."

Practice questions

Easy
  1. 1Is it possible to have a rectangle of perimeter 46 cm and area 135 cm²?
  2. 2Is there a real number x ≠ 0 such that x + 1/x = 4?
  3. 3Is it possible to have a rectangle of perimeter 50 cm and area 154 cm²?
Medium
  1. 4Is it possible to have a rectangle of perimeter 14 cm and area 14 cm²?
  2. 5Is there a real number x ≠ 0 such that x + 1/x = 0?
  3. 6Is it possible to have a rectangle of perimeter 52 cm and area 176 cm²?
Challenging
  1. 7Do two real numbers exist whose sum is 13 and product is 44?
  2. 8Do two real numbers exist whose sum is 12 and product is 47?
Show answers (try them yourself first)
  1. 1No
  2. 2Yes
  3. 3Yes
  4. 4No
  5. 5No
  6. 6No
  7. 7No
  8. 8No