Unit 2 - Functions
Required Prior Knowledge
Questions
Solve the equation \(2x^{2}+4x-5=0\) giving your solutions exactly.
Solutions
Get Ready (HL)
Questions
Using the same method, solve \(2x^{2}+4x+5=0\).
What do you notice? What do you wonder?
Solutions
Notes (HL)
Some quadratic equations have no real solutions.
But they can have complex solutions.
Your Turn
Find the complex solutions of the equation \(x^{2}+4=0\).
Your Turn
Determine the roots of the equation \(2x^{2}-6x+5=0\) in the form \(a+bi\).
Your Turn
The quadratic equation \(x^{2}+bx+c=0\) has complex roots \(1+i\) and \(1-i\).
Use this information to determine the constants \(b\) and \(c\).
Notes (SL)
There are many complicated looking equations which are actually a quadratic equation in disguise.
Look for the classic three terms that make up a quadratic to spot these.
Examples and Your Turns (SL)
Example
Solve the equation$$t^{4}-13t^{2}+36=0$$
Your Turn
Solve the equation$$2m^{4}-5m^{2}+2=0$$
Practice
Before attempting to answer each question, reflect on what has changed from the previous question.
What are your expectations for the answer?
Will it be bigger, smaller, the same?
How many answers will there be?
Then use the method to check.
Solve:
\(x^{2}-5x+6=0\)
\(x^{4}-5x^{2}+6=0\)
\(x^{4}-5x+6=0\)
\(x^{6}-5x^{3}+6=0\)
\(x^{8}-5x^{4}+6=0\)
\(x-5\sqrt{x}+6=0\)
\(x^{\frac{1}{2}}-5x^{\frac{1}{4}}+6=0\)
\(\left(x+1\right)^{2}-5\left(x+1\right)+6=0\)
\(\left(2x+1\right)^{4}-5\left(2x+1\right)^{2}+6=0\)
\(x^{\frac{2}{3}}-5x^{\frac{1}{3}}+6=0\)
\(2^{2x}-5\left(2^{x}\right)+6=0\)
\(\left(\log (2x)\right)^{2}-5\log (2x) +6=0\)
Give an example of an easy disguised quadratic equation.
Give an example of a difficult disguised quadratic equation.
Your Turn
Solve$$x^{4}-13x^{2}+36=0$$
Your Turn
Solve$$y^{6}-7x^{3}-8=0$$
Your Turn
Solve$$z^{\frac{2}{3}}-6x^{\frac{1}{3}}+7=0$$
Your Turn
Solve$$\left(x^{2}+3x\right)^{2}-2\left(x^{2}+3x\right)-8=0$$
Your Turn
Solve$$e^{2x}-3e^{x}+2=0$$
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
What does it mean for an equation to have complex solutions? How does this affect the graph?
Can a quadratic have one real and one complex root? Why or why not?
Describe how to ‘spot’ disguised quadratics.
Common Mistakes / Misconceptions
Misapplying the rules of complex numbers.
Not discarding solutions that can’t work for the original disguised quadratic.
Connecting This to Other Skills
We have obviously built upon the prior knowledge of solving quadratic equations (PK3) and also the basics of complext numbers (1.21).
Disguised quadratics will appear in Trigonometric Equations (3.16) and also in a variety of problems in other topics.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?