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?

 Jun 25, 2022

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



cool cool cool

 Jun 25, 2022

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