This week in Precalc 11 we learned about solving quadratic equations. While doing this I was severely overcomplicating how to solve for x. An example was when I was given the question:
x² – 3x + 2 = 0
When solving this I was not aware that you can just factor and solve for x by looking for what would make the equation equal to 0. Instead I was moving things around, switching signs and was very confused all around about what to do. To simplify what you can do is factor as usual …
x² – 3x + 2 = 0
= (x -2) and (x – 1)
And then when solving to find to make the equation equal to 0 just take x and figure out what would make (x -2) and (x – 1) equal to 0. Which is, 2 and 1.
This week in learning factoring, my favorite mistake was more of a silly mistake. When doing factoring I forgot that when doing a difference of square with two binomials that there is no middle term, and you can just solve with x² and the last term.
When solving radical equations for this unit, I had no idea that you were supposed to square both side if there was a square root attached to a number with x. Instead I was dividing by the square root and then trying to rationalize the denominator on the other side to make it “nice” and both sides even. I was overcomplicating it much more then I had to however because instead I could just square both sides which results in the radical on one side of the equal sign disappearing and just with a bigger number on the other side. Then for the most part you can just divide both sides of the equal sign by whatever coefficient is attached to the x.
Example:
√4x = 2
Square √4x and 2
(√4x) x (√4x) = 4x and 2 x 2 = 4
Divide both sides by the coefficient attached to x (4)
My mistake that I chose for this week was not rationalizing the denominator when simplifying a fraction under the radical symbol. When doing the question given instead of reducing the numbers if I could then rationalizing the denominator to get rid of the radical I simplified and put another fraction on the outside, getting the wrong answer with two fractions, one as a coefficient, one under the radical symbol. Instead I needed to reduce the fraction if I could and rationalize the denominator then multiply the numerator of the fraction by the denominator as well.
Example:
√27/10
I needed to reduce the √27 to 3√3 and then multiply that by √10 as well as the denominator receiving 10 because √10 x √10 = 10
then getting the final answer as 3/10√30 as my final answer
One mistake I made this week in the Adding and Subtracting Radicals unit was not knowing that with the same base underneath the radical sign you can just add or subtract the coefficients on the outside of the radical symbol, keeping the bas the same and get the right answer. I was under the impression I would have to convert the mixed radical to a entire radical to properly solve the equation, however I was overcomplicating it.
For example:
7√3 + 5√3 = 12√3
7√3 – 5√3 = 2√3
As shown above you can just add/subtract the coefficients/numbers in front of the radical sign when you have the same base under the radical symbol, and you get the right answer.
I chose to use determining the values of numbers with negative fractions for exponents. When I first encountered a question like this, because I was away the day we learned how to solve them, I thought I could just put the number and fractional exponent under 1 like you do with all negative exponents and be done with it, without finishing the rest of the equation. However I still needed to square root it by the denominator because it was 2 and it was easier because the numerator was 1, leaving it with the final answer being 1/3 simplified.
A mistake I made this week was not knowing that when multiplying two radicals you can just multiply them as is, right under the radicand. I originally overcomplicated the equations I was doing by square rooting the numbers under the radicand, then multiplying them and thinking that was the answer. Instead I should just multiply the numbers under the radicand and I could get the right answer. It’s important to know concepts such as multiplying and dividing radicals and the differences between how to add and subtract and multiply and divide.
