Consider the quadratic function with the equation in the general form $y = ax^2 + bx + c, \ a \neq 0$. Write down the formula, in terms of $a$, $b$ and $c$, of the solutions (roots) to the equation $ax^2 + bx + c = 0$.[1] 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] 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] Hence form inequalities for each of the following cases, in terms of $a$, $b$ and $c$, such that the graph of $y = ax^2 + bx + c$ is completely above the $x$-axis, the graph of $y = ax^2 + bx + c$ is completely below the $x$-axis, 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.

IP2 Class Test (May 2025) Bonus Question

IP 2

10 marks

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

Write down the formula, in terms of $a$, $b$ and $c$, of the solutions (roots) to the equation $ax^2 + bx + c = 0$.[1]
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]
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]
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.

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