SAMACHEER KALVI MATH SOLUTION FOR EXERCISE 3.3 PART 1

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Samacheer Kalvi Math Solution for Exercise 3.3 part 1

(1)
Find the zeros of the following quadratic polynomials and verify the
basic relationships between the zeros and the coefficients

(i) x² - 2 x - 8

Solution:

Let p(x) = x² - 2 x - 8 = (x - 4) (x + 2)

p(x) = 0 => (x - 4) (x + 2) = 0

x = 4 x = - 2

p(4) = 4² - 2 (4) - 8

= 16 - 8 - 8

= 16 - 16

= 0

p(-2) = (-2)² - 2 (-2) - 8

= 4 + 4 - 8

= 8 - 8

= 0

sum of zeroes = 4 + (-2) = 2

product of zeroes = 4 (-2) = -8

x² - 2 x - 8

ax² + bx + c

a = 1, b = -2 and c = -8

Sum of roots α + β = -b/a

α + β = -(-2)/1

α + β = 2

Product of roots α β = c/a

α β = -8/1

α β = -8

Thus the relationship verified

(ii) 4 x² - 4 x + 1

Solution:

Let p(x) = 4 x² - 4 x + 1 = (2x - 1) (2 x - 1)

p(x) = 0 => (2x - 1) (2 x - 1) = 0

2 x - 1 = 0

2 x = 1

x = 1/2 x = 1/2

p(1/2) = 4 (1/2)² - 4 (1/2) + 1

= 4/4 - 2 + 1

= 2 - 2

= 0

sum of zeroes = (1/2) + (1/2) = 1

product of zeroes = (1/2)(1/2) = 1/4

4 x² - 4 x + 1

ax² + bx + c

a = 4, b = -4 and c = 1

Sum of roots α + β = -b/a

α + β = -(-4)/4

α + β = 1

Product of roots α β = c/a

α β = 1/4

Thus the relationship verified

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