Unit 1: A more complex look on square roots!

This is an example of a more challenging problem, following the sqaure root method.

Step 1: 6(-x^2+1)+4(x^2-1)=18

My first step is writing my original equation. My goal is to solve this equation using the Square Root Method. I will have to solve it and simplify it, until the x is by itself. At the end I should have an answer for the value of x.

Step 2: -6x^2+6+4x^2-4=18

My next step is to distribute both sides of the addition sign. First I distributed 6(-x^2+1) and got -6x^2+6 and then I did the same to the 4(x^2-1). When I did that part of the problem I got 4x^2-4. With both of those equations, I combined them and the result was the equation above.

Step 3: -2x^2+2=18

In this step I had to start to break my equation down by combining like terms. I started off by identifying my like terms as -6x^2 & 4x^2 and 6 & -4. I combined the x terms by adding -6 and 4 which gave me -2. I then added 6 and -4 which gave me 2. Once I got my individual terms, I was able to combine them and form my simplified equation shown above.

Step 4: -2x^2-2x^2+2-2=18-2

In order for me to isolate the term connected to the x from the term without an x, I had to cancel out my 2 term. Since it was being added to the -2x^2, I had to do the opposite which in this case was subtraction in order to cancel them out. Once I cancelled them out, I was left with -2x^2=16.

Step 5: -2x^2÷-2=16÷-2

In this step I continued to work towards getting the x by itself by diving -2 on both sides. When I divided it by -2x^2, I ended up with x^2 and then when I divided 16 by -2 I was left with -8. From this point I rewrote my equation as x^2= -8.

Step 6: x^2= ±√4*-2

The next step was for me to try to cancel out the x^2 by getting the square root of it. With the square root method it is a rule, in order to get rid of a number squared you must get the square root of it. After getting the square root, I am left with x by itself. Since, I did that I have to then do the same to the other side to factor out the 8.Due to the fact that 8 is not a perfect square I had to then find a perfect square in 8 and I came to the conclusion that 4 is one and that in order to get -8, you have to multiply 4 by -2. After figuring that out I was left with the equation above.

Step 7: x= ±2i√2

Lastly, I had to simplify the terms inside of the radical. Knowing that 4 was a perfect square, I followed the rules of square roots and took the perfect square out of the radical and attached it to the outside along with the plus or minus sign. That then left me with 2√-2. I had to include the plus or minus sign because of the fact that it is the result of an even square root, so it can have both the negative and positive form of the number being squared to get that answer .Once, I got to that point I realized that there was a negative term in the radical so I used the rules of square roots to cancel it out so I took away the negative sign and attached an i to the 2 on the outside. After attaching the i, my simplified equation came out to be the equation above. I have successfully solved this problem, due to the fact that I have gotten two solutions for x.