if the sum of roots of ax²+bx+c=0 is =to the sum of the square of roots.find condition

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Please find below the solution to the asked query:

We have polynomial  f ( x ) = a x² + b x + c


Let zeroes are $\mathrm{\alpha }$  and $\mathrm{\beta }$
And
we know from relationship between zeros and coefficient . 

Sum of zeros  = $-\frac{\mathrm{Coefficient} \mathrm{of} \mathrm{x}}{\mathrm{Coefficient} \mathrm{of} {\mathrm{x}}^2}$
So,
$\mathrm{\alpha }$  + $\mathrm{\beta }$  =  $- \frac{\mathrm{b}}{\mathrm{a}}$                                                --- ( 1 )

And

Products of zeros  = $\frac{\mathrm{Const}an\mathrm{t} \mathrm{term}}{\mathrm{Coefficient} \mathrm{of} {\mathrm{x}}^2}$
So,
$\mathrm{\alpha }$  $\mathrm{\beta }$  =  $\frac{\mathrm{c}}{\mathrm{a}}$                                              --- ( 2 )

Now we take whole square of equation 1 and get

( $\mathrm{\alpha }$  + $\mathrm{\beta }$  )²  =  ${\left(- \frac{\mathrm{b}}{\mathrm{a}}\right)}^2$ 


$\mathrm{\alpha }$² + $\mathrm{\beta }$² + 2 $\mathrm{\alpha }$ $\mathrm{\beta }$ = $\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}$ , Now substitute value from equation 2 and get

$\mathrm{\alpha }$² + $\mathrm{\beta }$² + 2( $\frac{\mathrm{c}}{\mathrm{a}}$ ) = $\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}$

$\mathrm{\alpha }$² + $\mathrm{\beta }$² = $\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}$ - $\frac{2 \mathrm{c}}{\mathrm{a}}$

Given : Sum of roots and sum square of roots are equal , So

$- \frac{\mathrm{b}}{\mathrm{a}}$ = $\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}$ - $\frac{2 \mathrm{c}}{\mathrm{a}}$

$\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}$ - $\frac{2 \mathrm{c}}{\mathrm{a}}$ + $\frac{\mathrm{b}}{\mathrm{a}}$ = 0

$\frac{{\mathrm{b}}^2 - 2 \mathrm{a} \mathrm{c} + \mathrm{a} \mathrm{b} }{{\mathrm{a}}^2}$ =

b² - 2 ac + ab = 0 

b² = 2 ac - ab                                                                  ( Ans )


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