Question

zeros of the quadratic polynomial are
$(5 - 2 \sqrt{3)} (5 + 2 \sqrt{3} )$
foam a quadratic polynomial​

Answer 1

$\bold \star{ \underline{ \underline{ \textsf{ \textbf{ \color{red}{Gíven : - }}}}}}$

• Two Zeroes of a quadratic polynomial.

$\qquad \blacktriangleright \: { \sf{ \color{green}{ \alpha = (5 + 2 \sqrt{3} )}}}$

$\qquad \blacktriangleright \: { \sf{ \color{green}{ \beta = (5 - 2 \sqrt{3} )}}}$

$\bold \star{ \underline{ \underline{ \textsf{ \textbf{ \color{red}{To \: Find : - }}}}}}$

• A quadratic polynomial

$\circ \: \: { \underline{ \boxed{ \sf{ \color{purple}{Sum \: of \: Zeroes = \alpha + \beta }}}}}$

$\qquad \leadsto \: {= (5 + 2 \sqrt{3} ) +(5 - 2 \sqrt{3}) }$

$\qquad \leadsto \: {= (5 + \cancel{ 2 \sqrt{3} }) +(5 - \cancel{2 \sqrt{3}}) }$

$\qquad \leadsto \: {= (5 + 5)}$

$\qquad \leadsto \: { \underline{\fbox{ \pink{= 10}}}}$

$\circ \: \: { \underline{ \boxed{ \sf{ \color{purple}{ Product \: of \: Zeroes = \alpha \times \beta }}}}}$

$\qquad \leadsto \: {= (5 + 2 \sqrt{3} ) \times (5 - 2 \sqrt{3}) }$

$\qquad \leadsto \: { \sf{(a - b)(a - b) = {a}^{2} - {b}^{2} }}$

$\qquad \leadsto \: { = {5}^{2} - {(2 \sqrt{3}) \: }^{2} }$

$\qquad \leadsto \: { = 25 - (4 \times 3)}$

$\qquad \leadsto \: { = 25 - 12}$

$\qquad \leadsto \: { \underline{\fbox{ \pink{= 13}}}}$

• Now, standard form of quadratic polynomial is

$\qquad \star \: \: { \sf{ \color{green}{ a {x}^{2} + b(x) + (c) }}} \\ \sf{where \: a ≠0}$

• The required polynomial is

$\sf{ {x}^{2} - (sum \: of \: zeroes)x +(product \: of \: zeroes) }$

$\qquad \: {\sf { \color{red}{ = {x}^{2} - 10x + 13}}}$

• Hence the quadratic polynomial is

$\qquad \: {\sf { \color{red}{ = {x}^{2} - 10x + 13}}}$

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Answer 2

$\small \red\bigstar \: \: \bold \red{GIVEN :-}$

$\small \underline\bold \orange{TWO ~ZEROES~ OF~ A ~QUADRATIC ~POLYNOMIAL }$

$\bold \red\rightarrowtail \bold \green{a = (5 + 2 \sqrt{3}) }$

$\bold \red\rightarrowtail \bold \green{b = (5 - 2 \sqrt{3}) }$

$\bold \red\bigstar \: \: \bold \red{To \: \: Find :-}$

$\small \underline\bold \orange{A~QUADRATIC~POLYNOMIAL}$

$\small\bold \ \spadesuit \: \: \bold \red{SUM~OF~ZEROES=A+B}$

$\bold\dashrightarrow \bold \red{(5 + 2 \sqrt{3}) + (5 - 2 \sqrt{3}) }$

$\bold\dashrightarrow \bold \red{(5 + \cancel{2 \sqrt{3}} + (5 - \cancel{2 \sqrt{3} })}$

$\bold\dashrightarrow \bold \red{(5 + 5)}$

$\bold\dashrightarrow \fbox \pink{10}$

$\small\bold\spadesuit \: \bold \red{PRODUCT~OF~ZEROES \: \: \blue{a \times b}}$

$\bold\dashrightarrow \bold \red{(5 + 2 \sqrt{3} ) \times (5 - 2 \sqrt{3}) }$

$\bold\dashrightarrow \bold \red{(a - b)(a - b) = {a}^{2} - {b}^{2} }$

$\bold\dashrightarrow \bold \red{5 {}^{2} - (2 \sqrt{3} ) {}^{2} }$

$\bold\dashrightarrow \bold \red{25 - (4 \times 3)}$

$\bold\dashrightarrow \bold \red{25 = 12}$

$\bold\dashrightarrow \fbox \pink{13}$

$\small\bold\spadesuit \: \bold \red{NOW ~STANDARD ~FORM~ OF ~QUADRATIC ~POLYNOMIAL</p><p>}$

$\bold \pink \bigstar \: \bold \pink{Ax {}^{2} + B(x) + (c) }$

$\large\bold \blue{WHERE~ A \: \cancel = \: 0 }$

$\bold\spadesuit \: \bold \red{THE \: REQUIRED \: POLYNOMIAL \: IS }$

$\bold{X^2-(SUM~OF~ZEROES) X + (PRODUCT~OF~ZEROES) }$

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☢ HENCE THE QUADRATIC POLYNOMIAL IS :-

$\large \bold \red{x {}^{2} - 10x + 13 \: \: \checkmark }$