Verified answer

x=7

x + √x = 10

x + √x - 10 = 0

√x = (-1 + √41)/2

(reject (-1 - √41)/2 because √x needs to be positive.)

x = ((-1 + √41)/2)^2 = (42 - 2√41)/4 = (21 - √41)/2

6 < √41 < 7

-7 < -√41 < -6

14 < 21-√41 < 15

7 < (21-√41)/2 < 7.5

7 < x < 8

So 7 is too small and 8 is too big.

The largest integral value of x for which x + √x < 10 is x = 7.