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प्रश्न
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.
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उत्तर १
`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))`
उत्तर २
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`
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