Review question

# For what values of $k$ is $x^2 + 6kx +144$ always positive? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6526

## Suggestion

Write $x^2 + 6kx +144$ in the form $(x+p)^2+q$ and thus obtain expressions for $p$ and $q$ in terms of $k$.

Hence find the range of values for $k^2$ such that $x^2+6kx+144$ is positive for all values of $x$, and deduce the corresponding range of values for $k$.

When drawing the graph of $y = (x+p)^2+q$, what do $p$ and $q$ represent?

How does knowing $p$ and $q$ help us to determine whether the quadratic function is always positive?

Here is the graph of $y = x^2+6kx+144$.