Question

Find the zeroes of the quadratic polynomial `4x^2 - 4x + 1` and verify the relation between the zeroes and the coefficients. 

Find the zeroes of the polynomial 4x² − 4x + 1 and verify there rationship between the zeroes and the coefficients.

Answer 1

`4x^2 ˗ 4x + 1 = 0` 

`⇒ (2x^2)-2(2x)(1)+(1)^2=0` 

`⇒ (2x-1)^2=0`          [`∵ a^2-2ab+b^2=(a-b)^2`]

`⇒(2x-1)^2=0` 

`⇒x=1/2 or x=1/2` 

Sum of zeroes =`1/2+1/2=1=1/1=(("Coefficient of x "))/(("Coefficient of "x^2))` 

Product of zeroes`=1/2xx1/2=1/4 ("Constand term")/(("Coefficint of "x^2))`

 

Answer 2

Given, Polynomial is 4x² − 4x + 1   ...(i)

⇒ 4x² − 2x − 2x + 1

⇒ 2x (2x − 1) − 1(2x − 1)

⇒ (2x − 1) (2x − 1)

⇒ x = `1/2`

Hence, zeroes of a given Polynomial is x = `1/2`

On comparing equation (i) with ax² + bx + c = 0,

we get a = 4, b = − 4 and c = 1

Now, the sum of zeroes = `(-"b")/"a" = (-(-4))/4 = 1`

Product of zeroes = `"c"/"a" = 1/4`

which Matches with:

Sum of the zero = `1/2 + 1/2 = 2/2 = 1`

Product of the zero = `1/2 xx 1/2 = 1/4`