Question

EXERCISE - 3.3
Find the zeroes of the following quadratic polynomials and verify the relationship between
the zeroes and the coefficients.

(ii) 4s2-4s+1



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Answer 1

Answer :

The zeroes of the given polynomial are 1/2 and 1/2

Given :

The quadratic polynomial is :

• 4s² - 4s + 1

Task :

• To find out the zeroes of the given polynomial
• Also to verify the relationship between the zeroes and the coefficients .

Solution :

Splitting the middle term for factorization :

$\sf4 {s}^{2} - 4s + 1 \\ \\ \longrightarrow \sf4 {s}^{2} - 2s - 2s + 1 \\ \\ \longrightarrow \sf2s(2s - 1) - 1(2s - 1) \\ \\ \longrightarrow \sf(2s - 1)(2s - 1)$

The zeroes of the polynomial are :

$\sf2s - 1 = 0\: \: and \: \: 2s - 1 = 0 \\ \\ \implies \sf2s = 1 \: \: and \implies2s = 1 \\ \\ \implies \sf s = \dfrac{1}{2} \: \: and \implies s = \dfrac{1}{2}$

_______________________

Verification of the relationship between the zeroes and the coefficients :

$\sf{Sum \: \: of \: \: the \: \: roots = \dfrac{ -coefficient \: of \: x}{coefficient \: \: of \: \: {x}^{2} } }$

$\implies \sf \dfrac{1}{2} + \dfrac{1}{2} = \dfrac{ - ( - 4)}{4} \\ \\ \sf\implies \dfrac{1 + 1}{2} = \dfrac{4}{4} \\ \\ \implies \sf \dfrac{2}{2} = \dfrac{4}{4} \\ \\ \bf\implies1 = 1$

Again :

$\sf{Product \: \: of \: \: the \: roots \: = \dfrac{constant \: term}{coefficient \: \: of \: \: {x}^{2} } }$

$\sf \implies \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4} \\ \\ \bf \implies \dfrac{1}{4} = \dfrac{1}{4}$

$\bold{Hence \: \: Verified}$

Answer 2

$\huge\colorbox{orange}{♥︎ αɳʂɯҽɾ ♥︎} \\ \\ \\ \\ {\sf{\pink{∴ \: 4 {s}^{2} - 4s + 1 = 0}}} \\ {\sf{a {x}^{2} + bx + c = 0}} \\ {\sf{\blue{∴ \: a = 4, \: b = - 4, \: c = 1}}} \\ {\sf{\pink{∴ \: 4 {s}^{2} - 2s - 2s + 1 = 0}}} \\ {\sf{↬ \: 2s(2s - 1) - 1(2s - 1) = 0}} \\ {\sf{↬ \: (2s - 1)(2s - 1) = 0}} \\ {\boxed{\underline{\underline{\sf{\purple{∴s = \frac{1}{2} }}}}}} \\ \\ \\ {\underline{\sf{\green{Verification \: ✓}}}} \\ \\ {\boxed{\sf{\pink{sum \: of \: zeroes = \frac{ - b}{a} = \frac{ - ( - 4)}{4} = \frac{4}{4} = 1 }}}} \\ \\ {\boxed{\sf{\pink{product \:of \: zeroes = \frac{c}{a} = \frac{1}{4} }}}}$

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