Question

Find the zeroes of the quadratic polynomial √3x+-8x+4√3 and verify the relationship between the zeroes and the coefficients of the polynomial

Answer 1

$\large{\underline{\bf{\pink{Answer:-}}}}$

✰ x = 2√3 or x = 2/√3

$\large{\underline{\bf{\blue{Explanation:-}}}}$

$\large{\underline{\bf{\green{Given:-}}}}$

✰ f(x) = √3x² -8x+4√3

$\large{\underline{\bf{\green{To\:Find:-}}}}$

✰ we need to find the roots of the given polynomial. and also find the relationship between the zeroes and coefficients.

$\huge{\underline{\bf{\red{Solution:-}}}}$

f(x) = √3x² -8x+4√3

$:\implies\:\sf\sqrt{3}x^2-6x-2x+4\sqrt{3}$

$:\implies\:\sf\sqrt{3}x(x-2\sqrt{3})-2(x-2\sqrt{3})$

$:\implies\:\sf(x-2\sqrt{3})(\sqrt{3}x-2)$

$:\implies\:\sf(x-2\sqrt{3})=0 \:or\:(\sqrt{3}x- 2)=0$

$:\implies\:\sf\:x=2\sqrt{3}\:or\:x=\frac{2}{\sqrt{3}}$

So,

The zeroes of the given polynomial are 2√3 and 2/√3.

Sum of zeroes =$\sf2\sqrt{3}+\frac{2}{\sqrt3}=\frac{8}{\sqrt{3}}$

$:\implies\sf\frac{6+2}{\sqrt3}=\frac{8}{\sqrt{3}}$

$:\implies\sf{\purple{\frac{8}{\sqrt3}=\frac{8}{\sqrt{3}}}}$

product of zeroes =$\sf2\sqrt{3}\times\frac{2}{\sqrt3}=\frac{4\sqrt{3}}{\sqrt{3}}$

$:\implies\sf{\purple{\frac{4\sqrt{3}}{\sqrt{3}}=\frac{4\sqrt{3}}{\sqrt{3}}}}$

Hence,

$\bf\green{Relationship\: between \:the \:zeroes\: and}\:\\\bf\green {coefficients\: is\: \underline{verified}}$

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Answer 2

Correct Question :–

▪︎ Find the zeroes of the quadratic polynomial √3x² - 8x + 4√3 = 0 and verify the relationship between the zeroes and the coefficients of the polynomial.

ANSWER :–

$\\ \to \: \: { \bold{ Roots \: \: = 2 \sqrt{3} \: \: , \: \: \frac{2}{ \sqrt{3} } }}$

EXPLANATION :–

GIVEN :–

▪︎A quadratic equation √3x² - 8x + 4√3 = 0

TO FIND :–

▪︎ Zero's of quadratic equation.

▪︎To Verify the relationship between the zeroes and the coefficients of the polynomial.

SOLUTION :–

▪︎ According to the question –

$\\ \implies \: { \bold{ \sqrt{3} {x}^{2} - 8x + 4 \sqrt{3} = 0 }}$

$\\ \implies \: { \bold{ \sqrt{3} {x}^{2} - 6x - 2x + 4 \sqrt{3} = 0 }}$

$\\ \implies \: { \bold{ \sqrt{3}x(x- 2 \sqrt{3}) - 2(x - 2 \sqrt{3} ) = 0 }}$

$\\ \implies \: { \bold{ (\sqrt{3} x - 2)(x- 2 \sqrt{3}) = 0 }}$

$\\ \implies \: { \bold{ x = \dfrac{2}{ \sqrt{3} } \: , \: x = 2 \sqrt{3} }}$

▪︎ Now verification –

$\\ \: { \bold{ (1) \: sum \: \: of \: \: roots \: \: = - \frac{b}{a} }}$

$\\ \: { \bold{ \implies \: 2 \sqrt{3} + \frac{2}{ \sqrt{3} } = - \frac{( - 8)}{ \sqrt{3} } }}$

$\\ \: { \bold{ \implies \: \frac{6 + 2}{ \sqrt{3} } = \frac{ 8}{ \sqrt{3} } }}$

$\\ \: { \bold{ \implies \: \frac{8}{ \sqrt{3} } = \frac{ 8}{ \sqrt{3} } \: \: \: \: \: (verified)}}$

$\\ \: { \bold{ (2) \: Product \: \: of \: \: roots \: \: = \frac{c}{a} }}$

$\\ \implies \: { \bold{ \: (2 \sqrt{3} )( \frac{2}{ \sqrt{3} }) = \dfrac{4 \sqrt{3} }{ \sqrt{3} } }}$

$\\ \implies \: { \bold{ \: 4 = 4 \: \: \: \: \: \: (verified) }}$