Math, asked by NewBornTigerYT, 9 months ago

$\huge{\fbox{\fbox{\orange{\mathfrak{Explanation\:Required}}}}}$

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Answered by RvChaudharY50

Question :-

what must be added to (5x²/64) - {(3*√40*x)/20} + (22/25)

to make it a perfect Square ?

Solution :-

perfect Square is in the form of (a + b)² or (a - b)² .

Sign is negative in b/w, so lets take (a - b)²

→ (a - b)² = (a² - 2ab + b²)

→ (a² - 2ab + b²) = (5x²/64) - {(3*√40*x)/20} + (22/25)

comparing we get,

→ a² = (5x²/64)

Square - root both sides

→ a = (√5x/8)

comparing next,

→ 2ab = {(3*√40*x)/20}

→ 2ab = [ {3 * √(4*10)*x } / 20]

→ 2ab = [ { 2*3√10 * x } / 20 ]

cancel 2 from both

→ ab = (3√10*x) /20

putting value of a now,

→ (√5x/8) * b = (3 * √5 * √2 * x ) /20

cancel √5x from both sides ,

→ (b/8) = (3√2) / 20

→ b = (24√2/20)

→ b = (6√2/5)

Squaring both sides

→ b² = [(6√2)/5]²

→ b² = (36*2/25)

→ b² = (72/25)

so, now, we have 22/25 in Question ,

Adding part = (72/25) - (22/25) = (50/25) = 2 (Ans).

Hence, we can say That, if we add 2 in the given Algebraic Polynomial , we will get a perfect Square .

Also ,

it will be a perfect Square of = [ (√5x/8) - (6√2/5) ]² .

[ Nice Question. ]

Answered by Saby123

$\tt{\purple{\huge{Hello!!! }}}$ $\tt{\pink{------}}$ $\tt{\red{Question \: :- }}$ $\huge{\fbox{\fbox{\orange{\mathfrak{Explanation\:Required}}}}}$ $&lt;body bgcolor= "black"&gt;&lt;font color="yellow"&gt;$ :warning: No spam, if didn't answer etiquette answer then your id will be warned, may deleted too:arrow_right:Please answer, if you if felt this easy then definitely :ballot_box_with_check: check out my profile and kindly answer my questions $\tt{\blue{------}}$ $\tt{\blue{Solution \: :- }}$ We have to Factorise the above expression... $\tt{ \purple{ \leadsto{f(x) = \dfrac{5 {x}^{2} }{64} - \dfrac{3 \sqrt{40} x}{20} + \dfrac{22}{25} }}}$ Breaking as ( a- b )^2 , $\tt{ \orange{ \implies{ ({\dfrac{ \sqrt{5}x }{8} }^{2} ) - 2 \: \: \times \: {\dfrac{ \sqrt{5}x }{8} \: } \: \times \dfrac{3 \sqrt{5} }{5} + \dfrac{22}{5} }}}$ $\tt{\red{\leadsto{b =\dfrac{ 6 \sqrt{2}}{5} }}}$ $\tt{\green{\leadsto{ {b}^2 = \dfrac{72}{5}= \dfrac{22}{5} + 2 }}}$ Hence : 2 must be added to make it a perfect square. $\tt{\pink{------}}$

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