Question

Find the zeroes of the polynomial and find the relation between the zeroes and the coefficient.

4x^2-4x+1

Answer 1

$p(x) = 4 {x}^{2} - 4x + 1 \\ \\ p(x) = 4 {x}^{2} - 2x - 2x + 1 \\ \\p(x) = 2x(2x - 1) - 1(2x - 1) \\ \\p(x) = (2x - 1)(2x - 1)$

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Zeroes of the polynomial are :-

$2x - 1 = 0 \\ \\ x = \frac{1}{2}$

Zeroes are $\dfrac{1}{2}\:and\: \dfrac{1}{2}$

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☸️ Sum of zeroes :-

$\frac{1}{2} + \frac{1}{2} \\ \\ \\ = 1 \\ \\ = \frac{ - ( - 4)}{4} \\ \\ \\ = \frac{ - b}{a}$

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☸️ Product of zeroes:-

$\frac{1}{2} \times \frac{1}{2} \\ \\ = \frac{1}{4} \\ \\ \\ = \frac{c}{a}$

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$\boxed{ \boxed{ \red{ \sf{sum \: of \: zeroes = \frac{ - b}{a}}}}}$

$\rule{200}{2}$

$\boxed{ \boxed{ \orange{ \sf{product \: of \: zeroes = \frac{c}{a}}}}}$

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$\boxed{ \green{ \boxed{ \sf{verified}}}}$

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Answer 2

$\mathbb{\red{\huge{HEY\:MATE}}}$

$\boxed{HERE\:IS\:YOUR\:ANSWER}$

$\mathfrak{\green{ANSWER:}}$

Given,

Quadratic eq. = 4x² - 4x + 1

Here,

Coefficient of x² ( a ) = 4

Coefficient of x ( b ) = -4

Constant term ( c ) = 1

Now,

=> 4x² - 4x + 1 = 0

=> ( 2x )² - 2 × 2x × 1 + ( 1 )² = 0

Using identity :

=> ( a² - 2ab + b² ) = ( a - b )²

=> ( 2x - 1 )² = 0

=> ( 2x - 1 ) ( 2x - 1 ) = 0 ( continued further )

=> ( 2x - 1 ) = 0 ÷ ( 2x - 1 )

=> ( 2x - 1 ) = 0

=>$\huge\boxed{x = 1/2}$

=> ( 2x - 1 ) = 0 ÷ ( 2x - 1 )

=> ( 2x - 1 ) = 0

=> 2x = 1

=> $\huge\boxed{x = 1/2}$

Hence, zeroes are ( 1/2 ) and ( 1/2 ).

Now,

=> Sum of zeroes = -b/a

=> ( 1/2 ) + ( 1/2 ) = -( -4 ) ÷ 4

=> ( 1 + 1 )/2 = 4 ÷ 4

=> 2÷2 = 1

=> 1 = 1

And,

=> Product of zeroes = c/a

=> ( 1/2 ) × ( 1/2 ) = 1/4

=> 1/4 = 1/4

★ Verified ★