Math CentralQuandaries & Queries


Question from Roxanne:

Hello, i need to ask a question, do you mind explaining and writing the formula of how to solve equations such as "find the set values of k for which the line y=2x-5 cuts the curve y=x^2 +kx+11" please

Hi Roxanne,

I always draw pictures. Here is my picture of a line and a curve and a line where the line cuts the curve. I let $(a,b)$ be the point where the line cuts the curve.


Since $(a,b)$ lies on the line it satisfies the equation of the line and hence

\[b = 2a - 5.\]

The point $(a,b)$ is also on the curve so

\[b = a^2 + ka + 11.\]


\[2a - 5 = a^2 + ka + 11.\]

For what range of values of $k$ does the quadratic above have real roots?


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