EXERCISE 2.2
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
Sol. (i) We have, x2 – 2x – 8 = x2 – 2x – 4x-8
= x(x + 2) – 4(x + 2)
= (x + 2)(x – 4)
The value of x2 – 2x – 8 is zero when the value of (x + 2)(x – 4) is zero, i.e., when
x + 2 = 0 or x – 4 = 0, i.e.,
when x = – 2 or x = 4.
• The zeroes of x2 – 2x – 8 are – 2 and 4.
Therefore, sum of the zeroes = (-2) + 4 = 2
and, product of zeroes = (- 2)(4) = – 8 = 
(ii) We have, 4s2 – 4s + 1![]()
=-4s2 – 2s – 2s+1
= 2s(2s – 1) – 1(2s – 1)
= (2s – 1)(2s – 1)
The value of 4s2 – 4s + 1 is zero when the value of (2s – 1)(2s – 1) is zero, i.e.,
when 2s – 1 = 0 or 2s – 1 = 0,
i.e., when s = 
or s =
=The zeroes of 4s2 – 4s + 1 are and
Therefore, sum of the zeroes = + = 1
and, product of zeroes
(iii) We have,6x2 – 3 – 7x =6x2 – 7x – 3
= 6x2– 9x + 2x – 3
=3х(2x – 3) + 1(2x – 3)
=(3x + 1)(2x – 3)
The value of 6x2 – 3 – 7x is zero when the value of
(3x + 1)(2x – 3) is zero, i.e., when 3x + 1 = 0 or 2x – 3 = 0,
i.e., when x = – 
or x =
=> The zeroes of 6x2 – 3 – 7x are – and 
Therefore, sum of the zeroes
and, product of zeroes
(iv) We have, 4u2 +8u 4 = 4u(u + 2)
The value of 4u2 +8u is zero when the value of 4u(u + 2) is zero, i.e., when u = 0
or u + 2 = 0, i.e., when u = 0 or u = – 2.
=> The zeroes of 4u2 +8u are 0 and – 2.
sum of the zeroes = 0 ÷ (-2) =- 2
(v) We have, ![]()
t2 – 15
(vi) We have, 3x2 – x – 4 =3x2 -3 x – 4x- 4
= 3x(x + 1) – 4(x + 1)
= (x + 1)(3x – 4)
The value of 3x2 – x – 4 is zero when the value of (x + 1)(3x – 4) is zero, i.e.,
when x + 1 = 0 or 3x – 1 = 0, i.e.,
2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
Sol. (i) Let the polynomial be ax2 + bx + c, and its zeroes be a and 
. Then,
=> One quadratic polynomial which fits the given conditions is x2 – o.x + 
, i.e., x 2 + .
(iv) Let the polynomial be ax? + bx + c, and its zeroes be a and .
Then, a + = 1
a = 1
If a = 1, then b = -1and c = 1.
=> One quadratic polynomial which fits the given conditions is -x +1
(v)Let the polynomial be ax2 + bx + c and its zeroes be a and .
Then,
If a = 4, then b = -1and c = 1.
=> One quadratic polynomial which fits the given conditions is 4x2 -x + 1.
(vi) Let the polynomial be ax? + bx + c and its zeroes be a and B. Then,
If a = 4, then b = -4and c = 1.
=> One quadratic polynomial which fits the given conditions is x2 -4x + 1.