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Exercise - 2.2

Question-1 :-  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

Solution :-
(i) p(x) = x² – 2x – 8
         = x² – 4x + 2x – 8
         = x(x - 4) + 2(x - 4)
         = (x + 2)(x - 4)
    Zeroes of polynomial :
    x + 2 = 0
        x = -2
    x - 4 = 0
        x = 4
  
    Relationship between the zeroes and the coefficients :
    p(x) = x² – 2x – 8
    a = 1, b = -2, c = -8
    Sum of zeroes = α + β = -b/a = -(-2)/1 = 2
    Product of zeroes = αβ = c/a = -8/1 = -8
    
(ii) p(s) = 4s² – 4s + 1
         = 4s² – 2s - 2s + 1
         = 2s(2s - 1) - 1(2s - 1)
         = (2s - 1)(2s - 1)
    Zeroes of polynomial :
    2s - 1 = 0
        2s = 1
         s = 1/2, 1/2
  
    Relationship between the zeroes and the coefficients :
    p(s) = 4s² – 4s + 1
    a = 4, b = -4, c = 1
    Sum of zeroes = α + β = -b/a = -(-4)/4 = 1
    Product of zeroes = αβ = c/a = 1/4  
    
(iii) p(x) = 6x² – 3 – 7x
         = 6x² – 7x - 3
         = 6x² – 9x + 2x – 3
         = 3x(2x - 3) + 1(2x - 3)
         = (2x - 3)(3x + 1)
    Zeroes of polynomial :
    2x - 3 = 0
        2x = 3
         x = 3/2
    3x + 1 = 0
        3x = -1
         x = -1/3
  
    Relationship between the zeroes and the coefficients :
    p(x) = 6x² – 3 – 7x
    a = 6, b = -7, c = -3
    Sum of zeroes = α + β = -b/a = -(-7)/6 = 7/6 
    Product of zeroes = αβ = c/a = -3/6 = -1/2
    
     
(iv) p(u) = 4u² + 8u
          = 4u(u + 2)
    Zeroes of polynomial :
    4u = 0
     u = 0
    u + 2 = 0
     u = -2
    Relationship between the zeroes and the coefficients :
    p(u) = 4u² + 8u
    a = 4, b = 8, c = 0
    Sum of zeroes = α + β = -b/a = -(8)/4 = -2 
    Product of zeroes = αβ = c/a = 0/4 = 0 
    
(v) p(t) = t² – 15
         = t² - (√15)²
         = (t + √15)(t - √15)
    Zeroes of polynomial :
    t + √15 = 0
        t = -√15
    t - √15 = 0
        t = √15
    Relationship between the zeroes and the coefficients :
    p(t) = t² – 15
    a = 1, b = 0, c = -15
    Sum of zeroes = α + β = -b/a = 0/1 = 0 
    Product of zeroes = αβ = c/a = -15/1 = -15
    
(vi) p(x) = 3x² – x – 4
         = 3x² – 4x + 3x – 4
         = x(3x - 4) + 1(3x - 4)
         = (3x - 4)(x + 1)
    Zeroes of polynomial :
    3x - 4 = 0
        3x = 4
         x = 4/3
     x + 1 = 0
         x = -1
         
    Relationship between the zeroes and the coefficients :
    p(x) = 3x² – x – 4
    a = 3, b = -1, c = -4
    Sum of zeroes = α + β = -b/a = -(-1)/3 = 1/3 
    Product of zeroes = αβ = c/a = -4/3 
    

Question-2 :-  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.

Solution :-
(i) 1/4, -1 
    Given that :
    Sum of zeroes = α + β = -b/a = 1/4
    Product of zeroes = αβ = c/a = -4/4
    Therefore, a = 4, b = -1, c = -4. 
    In general form, p(x) = ax² + bx + c.
    So, quadratic polynomial is 4x² - x - 4.
    
(ii) √2, 1/3
    Given that :
    Sum of zeroes = α + β = -b/a = 3√2/3
    Product of zeroes = αβ = c/a = 1/3
    Therefore, a = 3, b = -3√2, c = 1. 
    In general form, p(x) = ax² + bx + c.
    So, quadratic polynomial is 3x² - 3√2x + 1.
    
(iii) 0, √5
    Given that :
    Sum of zeroes = α + β = -b/a = 0/1
    Product of zeroes = αβ = c/a = √5/1
    Therefore, a = 1, b = 0, c = √5. 
    In general form, p(x) = ax² + bx + c.
    So, quadratic polynomial is x² + √5.
    
(iv) 1, 1
    Given that :
    Sum of zeroes = α + β = -b/a = 1/1
    Product of zeroes = αβ = c/a = 1/1
    Therefore, a = 1, b = -1, c = 1. 
    In general form, p(x) = ax² + bx + c.
    So, quadratic polynomial is x² - x + 1.
    
(v) -1/4, 1/4
    Given that :
    Sum of zeroes = α + β = -b/a = -1/4
    Product of zeroes = αβ = c/a = 1/4
    Therefore, a = 4, b = 1, c = 1. 
    In general form, p(x) = ax² + bx + c.
    So, quadratic polynomial is 4x² + x + 1.
    
(vi) 4, 1
    Given that :
    Sum of zeroes = α + β = -b/a = 4/1
    Product of zeroes = αβ = c/a = 1/1
    Therefore, a = 1, b = -4, c = 1. 
    In general form, p(x) = ax² + bx + c.
    So, quadratic polynomial is x² - 4x + 1.
    
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