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  • If the roots of the quadratic equation  are equal ,

If the roots of the quadratic equation ( a-b)x^2 + ( b -c)x + ( c -a ) =0 are equal , prove that 2a = b+c

 

 

 

 
 
 
 
 

Answers (1)

Given ( a-b)x^2 + ( b -c)x + ( c -a ) =0 

To prove 

2a = b+ c 

proof : In Ax^ 2 + B x + C = 0  when roots are equal B^2 = 4 AC

For the above equation 

A = ( a-b) ; B = ( b-c) ; C = ( c-a ) \\\\ ( b-c) ^ 2 = 4 ( a-b) ( c-a) \\\\ b^2 + c ^ 2- 2bc = 4 [ ac - a ^2 - bc + ab ] \\\\ b ^ 2 + c ^ 2 - 2bc = 4ac - 4 a ^2 - 4 bc + 4 ab \\\\ 4a^ 2 + b^2 + c^2 - 2 b c + 4 bc - 4 ac - 4 ab = 0 \\\\ 4 a ^ 2 + b ^ 2 + c ^2 + 2 bc - 4ac -4ab \\\\

4a^2 + ( b+c) ^ 2 - 4a \times ( b+c) = 0 \\\\ ( 2a )^2 + ( b+c) ^ 2 - 2 \times 2a ( b+c) = 0 \\\\( 2a - ( b+c))^2 = 0 \\\\ 2a = b+c

Hence proved 

Posted by

Ravindra Pindel

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