y=x²+3x-5 is a parabola with gradient dy/dx=2x+3 (derivative).
The line y=2x+b is a tangent to the parabola at some point (X,X²+3X-5).
Since b is the y intercept, we need a line which lies below the tangent point and therefore can’t intersect the parabola. So if we find X, we can determine b.
To match the gradient of the line, which is 2, 2X+3=2, so X=-½.
Therefore y=0.25-1.5-5=-6.25. The tangent is at (-0.5,-6.25).
If the line passes through the tangent point, we can find b, using y=2x+b, and plugging in the tangent point:
-6.25=-1+b, so b=-5.25. This is the y intercept when the line is a tangent, so if b<-5.25 the line cannot intersect the parabola or touch it.
The picture shows the red tangent line when b=-5.25, and the blue line is one of the lines that doesn’t intersect the parabola because b<-5.25.
