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NCERT Class 10 Mathematics Exercise 4.3 Solutions: Quadratic Equations

Step-by-step working for every question of Exercise 4.3 in NCERT's Class 10 Mathematics book, chapter “Quadratic Equations”, written by Shishya. Each question is in our own short words; open the NCERT chapter for the exact wording and figures.

1. Ex 4.3 Q1

   Find the nature of the roots and the roots if real: (i) 2x² − 3x + 5 = 0 (ii) 3x² − 4√3x + 4 = 0 (iii) 2x² − 6x + 3 = 0.

   1. (i) For 2x² − 3x + 5 = 0, a = 2, b = −3, c = 5.
   2. (i) Discriminant D = b² − 4ac = (−3)² − 4(2)(5) = 9 − 40 = −31.
      why?

      For ax² + bx + c = 0 the discriminant D = b² − 4ac decides the roots: D > 0 two different real roots, D = 0 two equal real roots, D < 0 no real roots.

   3. (i) Since D < 0, the equation has no real roots.
   4. (ii) For 3x² − 4√3x + 4 = 0, a = 3, b = −4√3, c = 4.
   5. (ii) Discriminant D = b² − 4ac = (−4√3)² − 4(3)(4) = 48 − 48 = 0.
      why?

      For ax² + bx + c = 0 the discriminant D = b² − 4ac decides the roots: D > 0 two different real roots, D = 0 two equal real roots, D < 0 no real roots.

   6. (ii) Since D = 0, the equation has two equal real roots.
   7. (ii) Root x = −b/2a = 4√3 / 6 = 2√3 / 3 = 2/√3.
   8. (iii) For 2x² − 6x + 3 = 0, a = 2, b = −6, c = 3.
   9. (iii) Discriminant D = b² − 4ac = (−6)² − 4(2)(3) = 36 − 24 = 12.
      why?

      For ax² + bx + c = 0 the discriminant D = b² − 4ac decides the roots: D > 0 two different real roots, D = 0 two equal real roots, D < 0 no real roots.

   10. (iii) Since D > 0, the equation has two distinct real roots.
   11. (iii) Roots x = (6 ± √12) / 4 = (6 ± 2√3) / 4 = (3 ± √3) / 2.

   Answer: (i) no real roots; (ii) two equal roots, 2/√3; (iii) two distinct real roots, (3 + √3)/2, (3 - √3)/2

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2. Ex 4.3 Q2

   Find k so that each equation has two equal roots: (i) 2x² + kx + 3 = 0 (ii) kx(x − 2) + 6 = 0.

   1. (i) For 2x² + kx + 3 = 0, a = 2, b = k, c = 3.
   2. (i) Condition for equal roots is b² − 4ac = 0.
      why?

      For ax² + bx + c = 0 the discriminant D = b² − 4ac decides the roots: D > 0 two different real roots, D = 0 two equal real roots, D < 0 no real roots.

   3. (i) Substituting values gives k² − 4(2)(3) = 0.
      why?

      Putting the known values in place of the letters turns the formula or equation into plain arithmetic — the letters stood for exactly these values.

   4. (i) Simplifying yields k² − 24 = 0, so k² = 24.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   5. (i) Taking square root gives k = ±√24 = ±2√6.
   6. (ii) Expanding kx(x − 2) + 6 = 0 gives kx² − 2kx + 6 = 0.
      why?

      Expanding multiplies every term in one bracket by every term in the other: (a + b)(c + d) = ac + ad + bc + bd.

   7. (ii) Here a = k, b = −2k, c = 6.
   8. (ii) Condition for equal roots is b² − 4ac = 0.
      why?

      For ax² + bx + c = 0 the discriminant D = b² − 4ac decides the roots: D > 0 two different real roots, D = 0 two equal real roots, D < 0 no real roots.

   9. (ii) Substituting values gives (−2k)² − 4(k)(6) = 0.
      why?

      Putting the known values in place of the letters turns the formula or equation into plain arithmetic — the letters stood for exactly these values.

   10. (ii) Simplifying yields 4k² − 24k = 0.
       why?

       Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   11. (ii) Factoring gives 4k(k − 6) = 0.
   12. (ii) Since k ≠ 0 for a quadratic equation, k = 6.

   Answer: (i) k = ±2√6; (ii) k = 6

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3. Ex 4.3 Q3

   Can a rectangular mango grove have length twice its breadth and area 800 m²? If so, find its length and breadth.

   1. Let the breadth of the rectangular mango grove be x m.
   2. Since the length is twice the breadth, the length is 2x m.
   3. The area of a rectangle is given by length × breadth, so 2x × x = 800.
   4. Simplifying the equation gives 2x² = 800.
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   5. Dividing both sides by 2 gives x² = 400.
      why?

      Doing the same thing to both sides of an equation (adding, subtracting, multiplying or dividing by the same non-zero number) keeps the two sides equal, so the equation stays true while the unknown is left on its own.

   6. Taking the square root of both sides gives x = 20 (since breadth cannot be negative).
      why?

      If two non-negative quantities are equal, their square roots are equal. A square has two roots, + and −; a length or a count cannot be negative, so the negative root is dropped there.

   7. Therefore, the breadth is 20 m and the length is 2 × 20 = 40 m.

   Answer: yes; breadth 20 m; length 40 m

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4. Ex 4.3 Q4

   Two friends' ages add to 20 years and four years ago the product of their ages was 48. Is this possible? If so, find their ages.

   1. Let the present ages of the two friends be x and (20 − x) years.
   2. Four years ago, their ages were (x − 4) and (20 − x − 4), which simplifies to (x − 4) and (16 − x).
      why?

      Simplifying (combining like terms, cancelling common factors) changes how the expression looks, not its value, so the next step is easier.

   3. According to the problem, the product of their ages four years ago was 48.
   4. So, (x − 4)(16 − x) = 48.
   5. Expanding the left side gives 16x − x² − 64 + 4x = 48.
      why?

      Expanding multiplies every term in one bracket by every term in the other: (a + b)(c + d) = ac + ad + bc + bd.

   6. Combining like terms results in −x² + 20x − 64 = 48.
   7. Rearranging into standard quadratic form gives x² − 20x + 112 = 0.
   8. The discriminant is b² − 4ac = (−20)² − 4(1)(112) = 400 − 448 = −48.
      why?

      For ax² + bx + c = 0 the discriminant D = b² − 4ac decides the roots: D > 0 two different real roots, D = 0 two equal real roots, D < 0 no real roots.

   9. Since the discriminant is negative, the equation has no real roots.
      why?

      For ax² + bx + c = 0 the discriminant D = b² − 4ac decides the roots: D > 0 two different real roots, D = 0 two equal real roots, D < 0 no real roots.

   10. Therefore, it is not possible for the product of their ages four years ago to be 48.

   Answer: not possible

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5. Ex 4.3 Q5

   Can a rectangular park have perimeter 80 m and area 400 m²? If so, find its length and breadth.

   1. Let the length of the rectangular park be x m.
   2. Since the perimeter is 80 m, the sum of length and breadth is 80 ÷ 2 = 40 m.
   3. Therefore, the breadth of the park is (40 − x) m.
   4. The area of the park is given as 400 m², so x(40 − x) = 400.
   5. Expanding the equation gives 40x − x² = 400.
      why?

      Expanding multiplies every term in one bracket by every term in the other: (a + b)(c + d) = ac + ad + bc + bd.

   6. Rearranging the terms results in the quadratic equation x² − 40x + 400 = 0.
   7. The discriminant D is calculated as b² − 4ac = (−40)² − 4(1)(400) = 1600 − 1600 = 0.
      why?

      For ax² + bx + c = 0 the discriminant D = b² − 4ac decides the roots: D > 0 two different real roots, D = 0 two equal real roots, D < 0 no real roots.

   8. Since the discriminant is zero, the equation has two equal real roots, meaning the park can exist.
      why?

      For ax² + bx + c = 0 the discriminant D = b² − 4ac decides the roots: D > 0 two different real roots, D = 0 two equal real roots, D < 0 no real roots.

   9. The value of x is found using the formula −b ÷ 2a = 40 ÷ 2 = 20.
   10. Thus, the length is 20 m and the breadth is 40 − 20 = 20 m.

   Answer: yes, 20 m by 20 m

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