Question

Let $a\ne b$ and $2a^2=7a-5$, $2b^2=7b-5$ then find the value of $a^2+b^2$.

Answer 1

Given,

$a\ne b$ and $2a^2=7a-5$, $2b^2=7b-5$.

The above equations can be elaborated as $a,b$ are the roots of the equation $2x^2=7x-5$

or, $2x^2-7x+5=0$.

Then from the relation between the roots and co-efficient we get,

$a+b=-\frac{7}{2}$......(1) and $ab=\frac{5}{2}$......(2).

Now,

$a^2+b^2$

$=(a+b{)}^2-2ab$

$=\frac{49}{4}-2.\frac{5}{2}$ [ Using (1) and (2)]

$=\frac{29}{4}$

$=7\frac{1}{4}$.