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NCERT Exemplar · Class 10 · Maths

Chapter 4: Quadratic Equations

Practice each question, then open the answer or worked solution.

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Exercise 4.2

3 questions
Q1Written answer

State whether the following quadratic equations have two distinct real roots. Justify your answer.

  1. x2 – 3x + 4 = 0
  2. 2x2 + x – 1 = 0
  3. 2x2 – 6x + 9/2 = 0
  4. 3x2 – 4x + 1 = 0
  5. (x + 4)2 – 8x = 0
  6. (x – √2)2 – 2(x + 1) = 0
  7. √2 x2 –(3/√2)x + 1/√2 = 0
  8. x (1 – x) – 2 = 0
  9. (x – 1) (x + 2) + 2 = 0
  10. (x + 1) (x – 2) + x = 0
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Worked solution

(i)

The equation x2 – 3x + 4 = 0 has no real roots.

D = b2 – 4ac

= (-3)2 – 4(1)(4)

= 9 – 16 < 0

Hence, the roots are imaginary.

(ii)

The equation 2x2 + x – 1 = 0 has two real and distinct roots.

D = b2 – 4ac

= 12 – 4(2) (-1)

= 1 + 8 > 0

Hence, the roots are real and distinct.

(iii)

The equation 2x2 – 6x + (9/2) = 0 has real and equal roots.

D = b2 – 4ac

= (-6)2 – 4(2) (9/2)

= 36 – 36 = 0

Hence, the roots are real and equal.

(iv)

The equation 3x2 – 4x + 1 = 0 has two real and distinct roots.

D = b2 – 4ac

= (-4)2 – 4(3)(1)

= 16 – 12 > 0

Hence, the roots are real and distinct.

(v)

The equation (x + 4)2 – 8x = 0 has no real roots.

Simplifying the above equation,

x2 + 8x + 16 – 8x = 0

x2 + 16 = 0

D = b2 – 4ac

= (0) – 4(1) (16) < 0

Hence, the roots are imaginary.

(vi)

The equation (x – √2)2 – √2(x+1)=0 has two distinct and real roots.

Simplifying the above equation,

x2 – 2√2x + 2 – √2x – √2 = 0

x2 – √2(2+1)x + (2 – √2) = 0

x2 – 3√2x + (2 – √2) = 0

D = b2 – 4ac

= (– 3√2)2 – 4(1)(2 – √2)

= 18 – 8 + 4√2 > 0

Hence, the roots are real and distinct.

(vii)

The equation √2x2 – 3x/√2 + ½ = 0 has two real and distinct roots.

D = b2 – 4ac

= (- 3/√2)2 – 4(√2) (½)

= (9/2) – 2√2 > 0

Hence, the roots are real and distinct.

(viii)

The equation x (1 – x) – 2 = 0 has no real roots.

Simplifying the above equation,

x2 – x + 2 = 0

D = b2 – 4ac

= (-1)2 – 4(1)(2)

= 1 – 8 < 0

Hence, the roots are imaginary.

(ix)

The equation (x – 1) (x + 2) + 2 = 0 has two real and distinct roots.

Simplifying the above equation,

x2 – x + 2x – 2 + 2 = 0

x2 + x = 0

D = b2 – 4ac

= 12 – 4(1)(0)

= 1 – 0 > 0

Hence, the roots are real and distinct.

(x)

The equation (x + 1) (x – 2) + x = 0 has two real and distinct roots.

Simplifying the above equation,

x2 + x – 2x – 2 + x = 0

x2 – 2 = 0

D = b2 – 4ac

= (0)2 – 4(1) (-2)

= 0 + 8 > 0

Hence, the roots are real and distinct.

Q2Written answer

Write whether the following statements are true or false. Justify your answers.

  1. Every quadratic equation has exactly one root.
  2. Every quadratic equation has at least one real root.
  3. Every quadratic equation has at least two roots.
  4. Every quadratic equations has at most two roots.
  5. If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
  6. If the coefficient of x2 and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots.
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Worked solution

(i) False. For example, a quadratic equation x2 – 9 = 0 has two distinct roots – 3 and 3.

(ii) False. For example, equation x2 + 4 = 0 has no real root.

(iii) False. For example, a quadratic equation x2 – 4x + 4 = 0 has only one root which is 2.

(iv) True, because every quadratic polynomial has almost two roots.

(v) True, because in this case discriminant is always positive.

For example, in ax2+ bx + c = 0, as a and c have opposite sign, ac < 0

⟹ Discriminant = b2 – 4ac > 0.

(vi) True, because in this case discriminant is always negative.

For example, in ax2+ bx + c = 0, as b = 0, and a and c have same sign then ac > 0

⟹ Discriminant = b2 – 4ac = – 4 a c < 0

Q3Written answer

A quadratic equation with integral coefficient has integral roots. Justify your answer.

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Worked solution

No, a quadratic equation with integral coefficients may or may not have integral roots.

Justification

Consider the following equation,

8x2 – 2x – 1 = 0

The roots of the given equation are ½ and – ¼ which are not integers.

Hence, a quadratic equation with integral coefficient might or might not have integral roots.