Question

What is the largest positive integer value of m such that the equation

Answer 1

Using the discriminant from the quadratic formula (√b²-4ac) since that is the only way to get a solution to not be real (b²-4ac has to be negative), we get 
m²-4*3*7=m²-84. Solving for m if m²-84=0, m²=84 and m=+-√84. Therefore, if m is positive, the highest possible value of m is √84. However, we have to find the integer value - with a bit of guess and check, we find that 9²=81 and is the integer you get by rounding down √84 (if we rounded up, 10 would have real solutions since we need the square to be less than 84 to get m²-84 negative). Therefore, 9 is our answer