# For what values of k does the graph y=x^(2)+2kx+5 not intersect the x-axis

Where the graph intersects the x-axis, x^{2}+2kx+5 must be equal to zero. Thus we can answer the equivalent question: For what k does x^{2}+2kx+5 = 0 not have a solution?

This is now a simpler problem (roots of a quadratic equation). We can apply the common method of considering the discriminant of x^{2}+2kx+5. Using standard quadratic formula notation where in this case a=1, b =2k and c=5 we evaluate the discriminant : b^{2}-4ac= (2k)^{2}-4*1*5 = 4k^{2} -20.

Now since the discriminant appears in a square root sign in the quadratic equation, if it is negative there can be no real solutions to the equation ( great this is what we want!).

Thus we want discriminant negative: 4k^{2} -20 <0. Divide both sides of the inequality by 4 so we have k^{2}-5<0.

Now this is where we must take great care, the following reasoning is a common ** MISTAKE**: rearragne the inequality so we have k

^{2}< 5, then squarrot both sides so we have k < sqrt(5) or k < - sqrt(5) . The second inequalit is implied by the first thus the discriminate negtive for all k values les then the sqrt(5).

**.**

__THIS IS INCORRECT__When dealing with inequalities involving powers such as we are here we must be extremely careful. the mistake in the reasoning above is when we say k < - sqrt(5), this is actually a form of the common mistake of not inverting the inequality when multiplying both sides of an equation by a negative. Instead when dealing with inequalities with powers it is always much wiser to sketch a graph of the situation.

k^{2} - 5 is the standard quadratic U shape (think y=x^{2}) shifted down by 5. Having sketched this out it is clear that this graph is less then 0 when it is inbetween it's two roots.

The roots of k^{2 }-5 are easy to find: k^{2 }-5 = 0 implies k^{2} = 5 implies k = sqrt(5) or k = -sqrt(5).

Comparing this with the graph we can now see that the discriminant is negative for - sqrt(5) < k < sqrt(5). Thus these are the values for which the graph y=x^{2}+2kx+5 does not intersect the x-axis.