Math, asked by lvrkrajusri, 5 hours ago

If x+1/x=3 then xsquare+1/xsquare and x power 4+1/x power4​

Answers

Answered by Anonymous
36

Gɪᴠᴇɴ :-

 \:  \:  \:  \:  \:  \:  \:  \:  \tt \:  {x} +  \dfrac{1}{x}  = 3

Tᴏ Fɪɴᴅ :-

 \:  \:  \:  \bullet \:  \tt \: x {}^{2}  +  \dfrac{1}{x {}^{2} }  \\  \bullet \:  \tt \:   {x}^{4}  +    \frac{1}{x {}^{4} }

Sᴏʟᴜᴛɪᴏɴ :-

●Let's do squaring on both sides for given equation.

 \:  \:  \:  \:  \:  \:  \:  \:  \tt \:  \bigg( {x} +  \dfrac{1}{x}   \bigg) {}^{2} =( 3) {}^{2}

Using the formula,

 \boxed{ \underline{ \tt \:(a + b) {}^{2} = a {}^{2}  + 2ab + b {}^{2}   }}

 \tt \: x {}^{2}  +  \dfrac{1}{x {}^{2} }  + 2 \times x \times  \dfrac{1}{x}  = 9

 \tt \: x {}^{2}  +  \dfrac{1}{x {}^{2} }  + 2 \times  \not \: x \times  \dfrac{1}{ \not \: x}  = 9

 \:  \:  \tt \: x {}^{2}  +  \dfrac{1}{x {}^{2} }  + 2 = 9

 \:  \:  \tt \: x {}^{2}  +  \dfrac{1}{x {}^{2} }  = 9 - 2

 \boxed{   \red{\bigstar\:  \:  \tt  \: x {}^{2}  +  \dfrac{1}{x {}^{2} }  = 7}}

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Let's do squaring on both sides for what we got the value of x²+1/x² = 7

 \:  \:  \tt \: \bigg( x {}^{2}  +  \dfrac{1}{x {}^{2} }  \bigg) {}^{2}  = (7) {}^{2}

 \tt \: x {}^{4}  +  \dfrac{1}{x {}^{4} }  + 2 \times x {}^{2}  \times  \dfrac{1}{x {}^{2} }  = 49

 \tt \: x {}^{4}  +  \dfrac{1}{x {}^{4} }  + 2 \times  \not \: x  {}^{2} \times  \dfrac{1}{ \not \: x {}^{2} }  =4 9

 \:  \:  \tt \: x {}^{4}  +  \dfrac{1}{x {}^{4} }  + 2 = 49

 \:  \:  \tt \: x {}^{4}  +  \dfrac{1}{x {}^{4} }  = 49 - 2

 \boxed{   \red{\bigstar\:  \:  \tt  \: x {}^{4}  +  \dfrac{1}{x {}^{4} }  = 47}}

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Lᴇᴀʀɴ Mᴏʀᴇ :-

⇢( a +b )² = a² + 2ab + b²

⇢( a - b )² = a² + b² - 2ab

⇢( a + b )² + ( a - b)² = 2a² + 2b²

⇢( a + b )² - ( a - b)² = 4ab

⇢( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca

⇢a² + b² = ( a + b)² - 2ab

⇢(a + b )³ = a³ + b³ + 3ab ( a + b)

⇢( a - b)³ = a³ - b³ - 3ab ( a - b)

⇢If a + b + c = 0 then a³ + b³ + c³ = 3abc.

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