Unit 2 - Functions
Required Prior Knowledge
Questions
Solve the inequality$$2x^{2}-5x-3\gt 0$$
Solutions
Get Ready
Questions
By sketching graphs of quadratic functions, explain how it is possible for a quadratic function to have \(0\), \(1\) or \(2\) zeros.
Solutions
Notes
Given the quadratic equation in the form \(ax^{2}+bx+c=0\), where \(a\), \(b\), \(c\in\mathbb{R}\), the discriminant is the value of the expression$$\Delta =b^{2}-4ac$$Note that this is the bit of the quadratic formula inside the square root.
There are three possible cases:
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This means that we can evaluate the square root as a real number.
There are 2 distinct real roots.
This means the graph crosses the \(x\)-axis in two distinct points.
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This means that the square root evaluates to \(0\).
There is 1 repeated real root.
This means the graph sits on the \(x\)-axis at a single point.
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This means that we cannot evaluate the square root as a real number.
There are 0 real roots.
This means the graph never crosses the \(x\)-axis.
Note that this does not mean there are no roots. As we saw in 2.14 there are two complex roots.
As an extra, if \(\Delta\) is a square number, then the roots will be rational.
We can use these rules to discriminate between different types of quadratics by how many real roots they have.
Examples and Your Turns
Example
What can you deduce about the roots of these equations:
a) \(2x^{2}-3x-4=0\)
b) \(2x^{2}-3x-5=0\)
c) \(2x^{2}-4x+5=0\)
d) \(2x^{2}-4x+2=0\)
Your Turn
Determine the nature of the roots of:
a) \(2x^{2}-2x+3=0\)
b) \(3x^{2}-4x-2=0\)
Example
The equation \(kx^{2}-2x-7=0\) has two real roots. What can you deduce about the value of the constant \(k\)?
You can use the graph below to see what happens for different values of \(k\).
Your Turn
The equation \(3x^{2}+2x+k=0\) has a repeated root. Find the value of \(k\)?
You can use the graph below to see what happens for different values of \(k\).
Your Turn
For \(4x^{2}+kx+9=0\) determine the value(s) of \(k\) so that the equation has no real roots.
Your Turn
For the equation \(kx^{2}+\left(k+3\right)x=1\) determine the value(s) of \(k\) for which the equation has two distinct real roots.
Your Turn
For the function \(f\left(x\right)=3kx^{2}-\left(k+3\right)x+k-2\) determine the range of values of \(k\) for which \(f\) has no real roots.
Your Turn
The equation \(x^{2}+\left(p+1\right)x+\left(p+4\right)=0\) has no real roots. Find the possible values of \(p\).
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Explain how the results of the discriminant (\(\Delta\gt 0\), \(\Delta =0\) and \(\Delta\lt 0\)) relate to the graph of a quadratic function.
How can you use the discriminant to determine if you can easily factorise a quadratic?
Common Mistakes / Misconceptions
Forgetting to put brackets around negative values, especially when squaring. For example \(-5^{2}\) is not the same as \(\left(-5\right)^{2}\).
Incorrectly setting up the initial problem in terms of the inequality with the discriminant.
Forgetting that if \(ax^{2}+bx+c\) is a quadratic function, then \(a\ne 0\). This can mean you need to exclude some values of \(k\) as a special case.
Connecting This to Other Skills
This builds on the concept of solving quadratic equations (PK3).
To solve the exam type questions, you often need to solve a quadratic inequality (2.15) which involves understanding quadratic functions (2.7).
Intersecting Function (2.17) problems sometimes require the use of the discriminant.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?