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Question

Solve the given quadratic equation by using the formula method, a(x2+1)=x(a2+1)

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Solution

The given equation a(x2+1)=x(a2+1) can be simplified as shown below:

a(x2+1)=x(a2+1)ax2+a=x(a2+1)ax2x(a2+1)+a=0

Thequadratic equation ax2x(a2+1)+a=0 is in the form ax2+bx+c=0 where a=a,b=(a2+1) and c=a.
We know that the quadratic formula is x=b±b24ac2a, therefore, substitute a=a,b=(a2+1) and c=a in x=b±b24ac2a as follows:

x=b±b24ac2a=((a2+1))±((a2+1))2(4×a×a)2×a
=(a2+1)±a4+1+2a24a22a=(a2+1)±a4+12a22a
=(a2+1)±(a21)22a
=(a2+1)±(a21)2a

x=(a2+1)+(a21)2a=a2+1+a212a=2a22a=a,x=(a2+1)(a21)2a=a2+1a2+12a=22a=1a
Hence, x=1a or x=a.

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