The graph of the equation y=ax^2+bx-18 is completely below the x axis. If , a^2=49 what is the largest possible integral value of b?
If a^2= 49 then either a = 7 or a = -7
If a = 7, the parabola turns upward ....either the parbola lies entirely above the x axis or it intersects the x axis so it is impossible that a = 7
Then a = -7.....the parabola turns downward and lies completely below the x axis
So we have -7x^2 + bx - 18
Let's suppose that this parabola has just one root (its x coordinate of its vertex is the root)
Then the discriminant must = 0....so.....
b^2 - 4(-7)(-18) = 0
b^2 - 504 = 0
b^2 = 504
b ≈ 22.45
If b were larger than this we would have two roots and if b is smaller than this we have no real roots (the parabola lies completely below the x axis....what we want)
So.....the largest interger value for b that puts the parabola entirely below the x axis is when b= 22