Math, asked by MrMichael, 4 months ago

√7-2/√7+2=a√7+b
please solve this question​

Answers

Answered by LivetoLearn143
5

\large\underline{\sf{Solution-}}

It is provided that,

 \sf \: \dfrac{ \sqrt{7}  - 2}{ \sqrt{7} + 2 }  = a \sqrt{7}  + b

On rationalizing the denominator,

 \sf \: \dfrac{ \sqrt{7}  - 2}{ \sqrt{7} + 2 }   \times  \dfrac{ \sqrt{7}  - 2}{ \sqrt{7}  - 2} = a \sqrt{7}  + b

 \sf \: \dfrac{ (\sqrt{7}  - 2)^{2} }{ (\sqrt{7})^{2} -   {2}^{2}  }  = a \sqrt{7}  + b

We know that

 \sf \:  {(x  -  y)}^{2} =  {x}^{2} +  {y}^{2}  -  2xy

and

 \sf \: (x + y)(x - y) =  {x}^{2}  -  {y}^{2}

Therefore, we have

 \sf \: \dfrac{ 7 + 4 - 4 \sqrt{7} }{ 7 - 4  }  = a \sqrt{7}  + b

 \sf \: \dfrac{11 - 4 \sqrt{7} }{3}  = a \sqrt{7}  + b

 \sf \:\dfrac{11}{3}   - \dfrac{4 \sqrt{7} }{3}  = a \sqrt{7}  + b

So, if we compare,

 \sf \: b = \dfrac{11}{3}

 \sf \: a =  -  \: \dfrac{4}{3}

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