IP2 Class Test (May 2025) Bonus Question

Consider the quadratic function with the equation in the general form \(y = ax^2 + bx + c, \ a \neq 0\).

1. Write down the formula, in terms of \(a\), \(b\) and \(c\), of the solutions (roots) to the equation \(ax^2 + bx + c = 0\).[1]
2. It is given that the expression \(b^2 - 4ac\) is called the discriminant of the equation \(ax^2 + bx + c = 0\).
   State the relationship between the number of distinct real roots of the equation \(ax^2 + bx + c = 0\) and the discriminant.[3]
3. State the relationship between the number of distinct real roots of the equation \(ax^2 + bx + c = 0\) and the number of points of intersections between the graph of \(y = ax^2 + bx + c\) and the \(x\)-axis.[3]
4. Hence form inequalities for each of the following cases, in terms of \(a\), \(b\) and \(c\), such that
   1. the graph of \(y = ax^2 + bx + c\) is completely above the \(x\)-axis,
   2. the graph of \(y = ax^2 + bx + c\) is completely below the \(x\)-axis,
   3. the graph of \(y = ax^2 + bx + c\) and the \(x\)-axis have at least one common point.[3]

Note that \(a \neq 0\) throughout this question.