Question

The line y=2x+b is tangent to the graph of y= sqrt(x) at the point P = (a, sqrt(a) ). Find P and determine b.

Answer 1

\[2x+b=\sqrt{x}\]
\[(2x+b)^2=x\]
\[4x^2+4xb+b^2=x\]
\[4x^2+x(4b-1)+b^2=0\]
Discriminant.
\[b^2-4ac=0\]
\[(4b-1)^2-4(4)(b^2)=0\]
\[16b^2-8b+1-16b^2=0\]
\[8b=1\]
\[b=\frac{1}{8}\]