Question

Solve the equation \[{{\left( x-1 \right)}^{4}}+{{\left( x-5 \right)}^{4}}=82\]

Answer 1

Hint: In this question, we first need to write the given terms in the square terms to simplify it further. Then use the factorisation of polynomials and special products formula to expand it further. Now, expand all the terms and rearrange them to get the result.

Complete step-by-step answer:
Now, from the given equation in the question we have
\[\Rightarrow {{\left( x-1 \right)}^{4}}+{{\left( x-5 \right)}^{4}}=82\]
Now, this can also be written in square terms as
\[\Rightarrow {{\left( {{\left( x-1 \right)}^{2}} \right)}^{2}}+{{\left( {{\left( x-5 \right)}^{2}} \right)}^{2}}=82\]
As we already know that
\[{{x}^{2}}+{{a}^{2}}={{\left( x-a \right)}^{2}}-2ax\]
Now, by using this formula the above equation can be further written as
\[\Rightarrow {{\left( {{\left( x-1 \right)}^{2}}-{{\left( x-5 \right)}^{2}} \right)}^{2}}+2\times {{\left( x-1 \right)}^{2}}\times {{\left( x-5 \right)}^{2}}=82\]
As we already know the formula that
\[{{\left( x-a \right)}^{2}}={{x}^{2}}+{{a}^{2}}-2ax\]
Now, let us use the above formula and expand the terms
\[\Rightarrow {{\left( {{x}^{2}}+1-2x-\left( {{x}^{2}}+25-10x \right) \right)}^{2}}+2\times {{\left( x-1 \right)}^{2}}\times {{\left( x-5 \right)}^{2}}=82\]
Let us now further simplify it and rewrite it as
\[\Rightarrow {{\left( {{x}^{2}}+1-2x-{{x}^{2}}-25+10x \right)}^{2}}+2\times {{\left( x-1 \right)}^{2}}\times {{\left( x-5 \right)}^{2}}=82\]
Now, on further simplification we get,
\[\Rightarrow {{\left( 8x-24 \right)}^{2}}+2\times {{\left( x-1 \right)}^{2}}\times {{\left( x-5 \right)}^{2}}=82\]
Let us now expand the terms in the second part using the above mentioned formula
\[\Rightarrow {{\left( 8x-24 \right)}^{2}}+2\times \left( {{x}^{2}}+1-2x \right)\times \left( {{x}^{2}}+25-10x \right)=82\]
Now, on multiplying the respective terms we get,
\[\Rightarrow {{\left( 8x-24 \right)}^{2}}+2\times \left( {{x}^{4}}+25{{x}^{2}}-10{{x}^{3}}+{{x}^{2}}+25-10x-2{{x}^{3}}-50x+20{{x}^{2}} \right)=82\]
Now, on further simplification of the terms in the above equation and rearranging them we get,
\[\Rightarrow {{\left( 8x-24 \right)}^{2}}+2\times \left( {{x}^{4}}-12{{x}^{3}}+46{{x}^{2}}-60x+25 \right)=82\]
Let us now expand the terms in the first part of the above equation and simplify it further
\[\Rightarrow 64{{x}^{2}}+576-384x+2{{x}^{4}}-24{{x}^{3}}+92{{x}^{2}}-120x+50=82\]
Now, on further simplification we get,
\[\Rightarrow 2{{x}^{4}}-24{{x}^{3}}+156{{x}^{2}}-504x+626=82\]
Now, on rearranging the terms we get,
\[\therefore 2{{x}^{4}}-24{{x}^{3}}+156{{x}^{2}}-504x+544=0\]

Note: Instead of expressing them in terms of squares we can also simplify it by expressing it as a product of cube and linear and then expand it accordingly using the factorisation of polynomial formula. Both the methods give the same result.
It is important to note that while expanding the terms we should not neglect any of the terms or do calculation mistakes because it changes the corresponding equation and so the final result.