Find the zeroes of the quadratic polynomial x² – 2x – 8 and verify the relationship between the zeroes and the coefficients.

The polynomial is x² – 2x – 8 = (x – 4)(x + 2). The zeroes are 4 and -2. Sum of zeroes = 4 + (-2) = 2 = -(-2)/1 = -(coefficient of x)/(coefficient of x²). Product of zeroes = 4 × (-2) = -8 = -8/1 = (constant term)/(coefficient of x²). The relationship is verified.

Find the zeroes of the quadratic polynomial 6x² – 3 – 7x and verify the relationship between the zeroes and the coefficients.

Arranging the polynomial: 6x² – 7x – 3 = (3x + 1)(2x – 3). The zeroes are -1/3 and 3/2. Sum of zeroes = -1/3 + 3/2 = 7/6 = -(-7)/6 = -b/a. Product of zeroes = (-1/3) × (3/2) = -3/6 = c/a. The relationship is verified.

NCERT Solutions Class 10 Maths Chapter 2 Polynomials

Q: Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively: 1/4, – 1

Let the polynomial be ax² + bx + c. Sum of zeroes = -b/a = 1/4. Product of zeroes = c/a = -1. If a = 4, then b = -1 and c = -4. Therefore, the quadratic polynomial is 4x² – x – 4.

Q: Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively: √2, 1/3

Sum of zeroes = √2, Product of zeroes = 1/3. Using the formula k[x² – (sum)x + (product)], we get x² – √2x + 1/3. Multiplying by 3 to clear the denominator, the polynomial is 3x² – 3√2x + 1.

Detailed solutions for Exercise 2.2 (Relationship between Zeroes and Coefficients).

Exercise 2.2 • Q1

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x² - 2x - 8 (ii) 4s² - 4s + 1 (iii) 6x² - 3 - 7x
(iv) 4u² + 8u (v) t² - 15 (vi) 3x² - x - 4

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(i) x² - 2x - 8

Step 1: Find Zeroes
Split middle term: x² - 4x + 2x - 8
x(x - 4) + 2(x - 4)
(x - 4)(x + 2)
Zeroes are 4 and -2.

Step 2: Verify Relationship
Here a = 1, b = -2, c = -8
Sum of zeroes = 4 + (-2) = 2
-b/a = -(-2)/1 = 2 (Matched)

Product of zeroes = 4 × (-2) = -8
c/a = -8/1 = -8 (Matched)

(ii) 4s² - 4s + 1

Step 1: Find Zeroes
(2s - 1)² = 0
Zeroes are 1/2 and 1/2.

Step 2: Verification (a=4, b=-4, c=1)
Sum = 1/2 + 1/2 = 1
-b/a = -(-4)/4 = 1 (Verified)

Product = 1/2 × 1/2 = 1/4
c/a = 1/4 (Verified)

(iii) 6x² - 3 - 7x

Rearrange first: 6x² - 7x - 3
Step 1: Find Zeroes
6x² - 9x + 2x - 3
3x(2x - 3) + 1(2x - 3)
(3x + 1)(2x - 3)
Zeroes are -1/3 and 3/2.

Step 2: Verification (a=6, b=-7, c=-3)
Sum = -1/3 + 3/2 = (-2+9)/6 = 7/6
-b/a = -(-7)/6 = 7/6 (Verified)

Product = -1/3 × 3/2 = -3/6 = -1/2
c/a = -3/6 = -1/2 (Verified)

(iv) 4u² + 8u

Step 1: Find Zeroes
Take 4u common: 4u(u + 2)
Zeroes are 0 and -2.

Step 2: Verification (a=4, b=8, c=0)
Sum = 0 + (-2) = -2
-b/a = -8/4 = -2 (Verified)

Product = 0 × (-2) = 0
c/a = 0/4 = 0 (Verified)

(v) t² - 15

Step 1: Find Zeroes
Using a² - b² = (a-b)(a+b)
t² - (√15)² = (t - √15)(t + √15)
Zeroes are √15 and -√15.

Step 2: Verification (a=1, b=0, c=-15)
Sum = √15 - √15 = 0
-b/a = -0/1 = 0 (Verified)

Product = √15 × -√15 = -15
c/a = -15/1 = -15 (Verified)

(vi) 3x² - x - 4

Step 1: Find Zeroes
3x² - 4x + 3x - 4
x(3x - 4) + 1(3x - 4)
(x + 1)(3x - 4)
Zeroes are -1 and 4/3.

Step 2: Verification (a=3, b=-1, c=-4)
Sum = -1 + 4/3 = 1/3
-b/a = -(-1)/3 = 1/3 (Verified)

Product = -1 × 4/3 = -4/3
c/a = -4/3 (Verified)

Exercise 2.2 • Q2

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) 1/4, -1 (ii) √2, 1/3 (iii) 0, √5
(iv) 1, 1 (v) -1/4, 1/4 (vi) 4, 1

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Formula Used

Quadratic Polynomial = k [x² - (Sum of zeroes)x + (Product of zeroes)]

(i) Sum = 1/4, Product = -1

x² - (1/4)x + (-1)
x² - x/4 - 1
To remove fraction, multiply by 4:
Answer: 4x² - x - 4

(ii) Sum = √2, Product = 1/3

x² - (√2)x + (1/3)
Multiply by 3 to remove fraction:
Answer: 3x² - 3√2x + 1

(iii) Sum = 0, Product = √5

x² - (0)x + √5
Answer: x² + √5

(iv) Sum = 1, Product = 1

x² - (1)x + 1
Answer: x² - x + 1

(v) Sum = -1/4, Product = 1/4

x² - (-1/4)x + 1/4
x² + x/4 + 1/4
Multiply by 4:
Answer: 4x² + x + 1

(vi) Sum = 4, Product = 1

x² - (4)x + 1
Answer: x² - 4x + 1

NCERT Solutions Class 10 Maths Chapter 2: Polynomials

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x² - 2x - 8 (ii) 4s² - 4s + 1 (iii) 6x² - 3 - 7x
(iv) 4u² + 8u (v) t² - 15 (vi) 3x² - x - 4

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Hate verifying coefficients?

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) 1/4, -1 (ii) √2, 1/3 (iii) 0, √5
(iv) 1, 1 (v) -1/4, 1/4 (vi) 4, 1

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NCERT Solutions Class 10 Maths Chapter 2: Polynomials

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) x² - 2x - 8 (ii) 4s² - 4s + 1 (iii) 6x² - 3 - 7x (iv) 4u² + 8u (v) t² - 15 (vi) 3x² - x - 4

Stuck on this Question?

Hate verifying coefficients?

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i) 1/4, -1 (ii) √2, 1/3 (iii) 0, √5 (iv) 1, 1 (v) -1/4, 1/4 (vi) 4, 1

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Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x² - 2x - 8 (ii) 4s² - 4s + 1 (iii) 6x² - 3 - 7x
(iv) 4u² + 8u (v) t² - 15 (vi) 3x² - x - 4

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) 1/4, -1 (ii) √2, 1/3 (iii) 0, √5
(iv) 1, 1 (v) -1/4, 1/4 (vi) 4, 1