AMO 2027 Grades 10-12 SET 2 - Self-study Question 2

A quadratic equation of the form $f(x)=x^2+bx+c$ opens upwards and has its vertex 5 units above the $x$-axis and 7 units to the right of the $y$-axis. If a linear equation $g(x)=mx+10$ were added to the original equation to give $h(x)=f(x)+g(x)$, find the $x$-coordinate of the vertex if the graph of $h(x)$ just touches the positive $x$-axis.

Choose one option.