Q1 of 227 Page 82

Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.

4x2 – 4x + 1

Let f(x) = 4x2 – 4x + 1

To find out zeros of the given polynomial.


We put f(x) = 0


⇒ 4x2 – 4x + 1 = 0


To find out roots of this polynomial we use splitting of middle term method.


According to this method we need to find two numbers whose sum is – 4 and product is 4.


∴ 4x2 – (2 + 2)x + 1 = 0


⇒ 4x2 – 2x – 2x + 1 = 0


⇒ 2x(2x – 1) – 1(2x – 1) = 0


⇒ (2x – 1)(2x – 1) = 0


∴ 2x – 1 = 0


∴ x = .


Again, 2x – 1 = 0


∴ x =


⇒ Our zeros are α = and β = .


⇒ sum of zeros = α + β = + = 1.


⇒ Product of zeros = αβ = .


Now, Comparing f(x) = 4x2 – 4x + 1 with standard equation ax2 + bx + c = 0.


We get, a = 4, b = – 4 and c = 1


We can verify,


⇒ Sum of zeros =


i.e. α + β =


∴ α + β = 1


⇒ Product of zeros =


αβ =


Hence, relationship between zeros and coefficient is verified.


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