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Questions
Find the zeroes of the quadratic polynomial 4x2 – 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.
Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the coefficients:
4x2 – 4x + 1
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Solution 1
4x2 – 4x + 1 = 0
⇒ (2x2) – 2(2x)(1) + (1)2 = 0
⇒ (2x – 1)2 = 0 ...[∵ a2 – 2ab + b2 = (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 = `1/2 xx 1/2`
= `1/4`
= `("Constant term")/(("Coefficient of "x^2))`
Solution 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`
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