Question

Kindly Solve it with a proper explanation :-)

#No spams :)


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Answer 1

Before we solve the problem

Sure, we may find the value of the given fraction.

But we can see that

$\sf{\dfrac{\cancel{(m-3)^4}}{\cancel{(m-3)^2}} +\dfrac{1}{(m-3)^2} }$

$\sf{=(m-3)^2+\dfrac{1}{(m-3)^2} }$

After transposing sides from the given

$\sf{(m-3)-\dfrac{1}{(m-3)} =1}$

And we approach much faster.

Let's get to the problem

After squaring and transposing sides

$\sf{(m-3)^2+\dfrac{1}{(m-3)^2} =1+2}$

The Conclusion

The value is 3.

For your information.

The concepts used:

• Identity
• Transposing the Sides

As we see from the structure.

We should not solve the equation,

since the question requires us to use identity.

Answer 2

Here,

$=>m - \frac{1}{ m- 3} = 4 \\ \\ => \frac{m(m - 3) - 1}{m - 3} = 4 \\=> {m}^{2} - 3m - 1 = 4m - 12 \\ =>{m}^{2} - 7m + 11 = 0 \\ \\=> \bigg(m - 3.5 + \frac{ \sqrt{5} }{2} \bigg) \bigg(m - 3.5 - \frac{ \sqrt{5} }{2}\bigg )$

From Above,

$\rm{}m = 3.5 + \frac{ \sqrt{5} }{2} \: \: \: \: \: \: or \: \: \: \: \: 3.5 - \frac{ \sqrt{5} }{2}$

Now,

$\frac{ {(m - 3)}^{4} + 1}{ {( m- 3)}^{2} } \\ \\ \frac{ {(m - 3)}^{4} }{ {( m- 3)}^{2} } + \frac{1}{ {(m - 3)}^{2} } \\ \\ {( m- 3)}^{2} + \frac{1}{ {(m - 3)}^{2} }$

For above,

$m - 3 = > 3.5 - \frac{ \sqrt{5} }{2} - 3 \\ = > 0.5 - \frac{ \sqrt{5} }{2} \\ \\ \\ m - 3 = > 3.5 + \frac{ \sqrt{5} }{2} - 3 \\ = > 0.5 + \frac{ \sqrt{5} }{2}$

Also,

$\rm {(m - 3)}^{2} = > { \bigg(0.5 - \frac{ \sqrt{5} }{2} \bigg)}^{2} \\ \\ = > 0.25 + \frac{5}{4} - 0.5 \sqrt{5} \\ \\ = > \frac{1 + 5 - 2 \sqrt{5}\} }{4} \\ \\ = > \frac{ 6 - 2 \sqrt{5} }{4} = \frac{2(3 - \sqrt{5} )}{4} \\ \\ \rm {(m - 3)}^{2} = > \frac{3 - \sqrt{5} }{2}$

Thus,

$\rm {(m - 3)}^{2} + \frac{1}{ {(m - 3)}^{2} } \\ \\ = > { \bigg( \frac{3 - \sqrt{5} }{2} \bigg)}^{} + \frac{1}{ {\bigg( \frac{3 - \sqrt{5} }{2} \bigg)}^{} } \\ \\ = > \frac{3 - \sqrt{5} }{2} + \frac{2}{3 - \sqrt{5} } \\ \\ = > \frac{3 - \sqrt{5} }{2} + \bigg \{ \frac{2(3 + \sqrt{5} }{(3 - \sqrt{5} )(3 + \sqrt{5} ) } \bigg\} \\ \\ = > \frac{3 - \sqrt{5} }{2} + \frac{6 + 2 \sqrt{5} }{ {3}^{2} - { (\sqrt{5} )}^{2} } \\ \\ = > \frac{3 - \sqrt{5} }{2} + \frac{2(3 + \sqrt{5} )}{9 - 5} \\ \\ = > \frac{3 - \sqrt{5} }{2} + \frac{2(3 + \sqrt{5}) }{4} \\ \\ = > \frac{3 - \sqrt{5} }{2} + \frac{3 + \sqrt{5} }{2} \\ \\ = > \frac{3 - \cancel{\sqrt{5}} + 3 + \cancel {\sqrt{5} } }{2} \\ \\ = > \frac{6}{2} \: \: \: = \red {3}$

Hence,

• The required value is 3