Below are several statements about the quadratic equation \[ax^2 + bx + c = 0,\] where \(a\), \(b\) and \(c\) are allowed to be any real numbers except that \(a\) is not \(0\).

For each statement, decide whether MUST, MAY or CAN’T is the correct word to use in the statement.

Do you have any favourite examples of quadratics with different numbers of real roots? Having some examples to test is a good way to begin on a problem like this.

How can we use the coefficients to determine the number of real roots?

How can thinking about the graph of the quadratic \(y=ax^2+bx+c\) help us to think about some of these statements?

As well as trying to visualise this, it may be helpful to use the interactive graph.