30.2k views
3 votes
Hello, may I ask how to find the imaginary solutions to this problem

Hello, may I ask how to find the imaginary solutions to this problem-example-1

1 Answer

3 votes

Hello!

We have the equation:


(1)/(2)x^2-x+5=0

Let's solve it by the method of completing the square:

I will put the unknowns on one side and the value on the other, look:


(1)/(2)x^2-x=-5

To remove the fraction, we can divide both sides by 1/2, obtaining:


x^2-2x=-10

Now let's leave space to complete the square:


x^2-2x+\square=-10+\square

The value that will be added must be the same on both sides.

Remember we want to complete a square, so, let's write this expression as a product of factors:


x^2-2\cdot x\cdot1+\square^2

Let's replace where is the square by 1 and solve this expression:


\begin{gathered} x^2-2\cdot x\cdot1+1^2 \\ (x^2-2x+1)\text{ we can write it as} \\ \mleft(x-1\mright)^2 \end{gathered}

Notice that we can get a square on the left side when we use 1. So let's replace the square on the right side with 1 as well:


\begin{gathered} (x-1)^2=-10+\square \\ (x-1)^2=-10+1 \\ (x-1)^2=-9 \end{gathered}

To solve this expression, we can apply the square root of both sides:


\begin{gathered} \sqrt[]{(x-1)^2}=\sqrt[]{-9} \\ x-1=\sqrt[]{-9} \end{gathered}

Now that the imaginary numbers part comes in, the square root of a negative number just exists in the imaginary numbers.

Let's calculate the square root of -9:

Remember that √9 = +3 or-3.

In the same way, to calculate the square root of a negative number we will follow the same steps and then replace the result with "i", in reference to imaginary numbers.

Knowing it let's finish your exercise:


\begin{gathered} x-1=\pm\sqrt[]{-9} \\ x-1=\pm3i \\ x=+1\pm3i \end{gathered}

As we can have a positive and a negative result, let's divide it into two results:


\begin{gathered} x_1=1+3i \\ x_2=1-3i \end{gathered}

User John Redyns
by
8.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories