SOLUTION: Let f(a,b) = 2a - 3b^2 + 7 + 5a^2 - 10a. If f(k,-3) = -10, then what is k?

Combine like terms

    f(a,b) = 5a^2 -3b^2 - 8a + 7.


Substitute a= k, b= -3

    f(k,-3) = 5k^2 - 3*(-3)^2 - 8k + 7.


Simplify

    f(k,-3) = 5k^2 -27 - 8k + 7 = 5k^2 - 8k - 20.


Your equation to find "k" is

    5k^2 - 8k - 20 = -10.


Simplify

    5k^2 - 8k - 10 = 0.


The discriminant is  d = (-8)^2 - 4*5*(-10) = 64 + 200 = 264.


Find "k" using the quadratic formula

     =  =  = 


ANSWER.  There are two real values for "k" :    and  .