14x squared plus 37x plus 5 => 14x^2 + 37x + 5
Think of the most basic parabola: y=x^2. This curve opens up and has its vertex at (0,0). Adding 5 to y=x^2 shifts the entire graph upward by 5 units; we call this "vertical translation." Going back to y = 14x^2 + 37x + 5, we can easily conclude that 5 also results in vertical translation of the graph of y=14x^2 + 37x. Whether or not the function y = 14x^2 + 37x + k has real roots or not depends upon the value of the constant k and can most easily be determined by calculating the discriminant b^2-4ac.
Here a=14, b=37 and c=5.
The discriminant is 37^2-4(14)(5) = 1089. Because this is positive, the function has 2 real, unequal roots. If we were to use some constant other than 5, then the discriminant would be 37^2-4(14)(k). Set this equal to 0 and solve for k: 37^2/(4*14) = k, so that k = 24.45. At this value of k, the function would have 2 real, equal roots. If we pick a value for k such that the discriminant is negative, then the function would have 2 complex roots.
