How to Avoid Extraneous Solutions in Solving Complex Equations

In summary, the conversation discusses solving a complex equation involving the absolute value of a complex number. One person tried to solve it using a squared equation, but they were reminded that this was incorrect. Instead, they were advised to rearrange the equation and solve for the imaginary part to be zero. The person also mentioned getting two solutions, but one was incorrect due to an extraneous solution introduced by squaring. It was suggested to check if the equation is valid with each solution to determine which one to discard.
  • #1
Yankel
395
0
Hello all,

Please look at the following:

Solve the equation:

\[\left | z \right |i+2z=\sqrt{3}\]

where z is a complex number.

I tried solving it, and did the following, which is for some reason wrong. I saw a correct solution. My question to you is why mine is not, i.e., where is my mistake ?

\[i\sqrt{x^{2}+y^{2}}+(2x+2iy)=\sqrt{3}\]

\[(x^{2}+y^{2})(-1)+(4x^{2}+8xiy-4y^{2})=3\]

\[3x^{2}-5y^{2}+8xiy=3\]

\[(1,0),(-1,0)\]

This is definitely wrong. Can you please tell me where my mistake it ?

Thank you !

The correct answer should be: \[\frac{\sqrt{3}}{2}-\frac{1}{2}i\]
 
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  • #2
Yankel said:
I tried solving it, and did the following, which is for some reason wrong. I saw a correct solution. My question to you is why mine is not, i.e., where is my mistake ?

\[i\sqrt{x^{2}+y^{2}}+(2x+2iy)=\sqrt{3}\]

\[(x^{2}+y^{2})(-1)+(4x^{2}+8xiy-4y^{2})=3\]

Hey Yankel,

You've squared the equation.
However, the left side was not squared correctly.
Note that $(a+b)^2 \ne a^2+b^2$.

Instead, there is no need to square at all.
We can rearrange the equation as:
\[2x + i \left(\sqrt{x^{2}+y^{2}}+2y\right)=\sqrt{3}\]
If follows directly that $x=\frac{\sqrt 3}2$, after which we can solve for the imaginary part to be zero.
 
  • #3
Thank you ! Silly mistake (Doh)

Solving your way, I get two solutions (y=1/2 and y=-1/2). One is incorrect. How can I know to ignore it without checking if the equation is valid with each solution ?
 
  • #4
Yankel said:
Solving your way, I get two solutions (y=1/2 and y=-1/2). One is incorrect. How can I know to ignore it without checking if the equation is valid with each solution ?

You would have squared to solve the imaginary part to be zero.
That introduces an extraneous solution.
Check just before squaring whether y is supposed to be positive or negative. Then we can tell after (or during) solving which one to discard.
 

FAQ: How to Avoid Extraneous Solutions in Solving Complex Equations

1. How do I approach solving a complex equation?

When faced with a complex equation, the first step is to simplify it as much as possible by combining like terms and using basic algebraic principles. Then, try to isolate the variable you are solving for on one side of the equation. If there are multiple variables, consider using substitution or elimination methods. It may also be helpful to graph the equation to get a visual understanding of the solution.

2. What should I do if I get stuck while solving a complex equation?

If you get stuck while solving a complex equation, take a step back and try to identify the specific part that is causing difficulty. Review any relevant mathematical principles or techniques that may help in solving that part. It may also be beneficial to seek help from a teacher or tutor.

3. Can I use a calculator to solve a complex equation?

Yes, calculators can be useful tools for solving complex equations, but they should not be solely relied upon. It is important to have a strong understanding of the underlying principles of algebra and not just rely on the calculator's solutions. Also, not all calculators are capable of solving every type of complex equation, so it is important to check the capabilities of your specific calculator.

4. How do I know if my solution is correct?

Once you have solved a complex equation, you can check your solution by plugging it back into the original equation and seeing if it satisfies the equation. You can also graph the equation and see if your solution falls on the graph. Another option is to use an online equation solver or ask someone else to check your work.

5. Are there any shortcuts or tricks for solving complex equations?

While there are some common techniques and patterns that can make solving certain types of complex equations easier, there are no universal shortcuts or tricks that apply to all equations. The best approach is to have a solid understanding of algebraic principles and to practice solving a variety of equations to develop problem-solving skills.

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