Week 8 in Precalc 11 – Quadratic formula

This week in Precalc 11, in addition to reviewing our past units, we learned about the Quadratic formula. The quadratic formula allows us to solve any quadratic equation that’s in the form ax^2 + bx + c = 0. This is useful because you can use this formula anytime you don’t want to do grouping, inspection, etc.

File:Quadratic formula.svg - Wikipedia

In an equation, this is where the ABCs are:

To solve an equation you simply plug in the numbers and solve:

In the end, if you are left with a radical that can be simplified, make sure to simplify. Then you work out the equations till you get your answers.

In this example I have simplified the radical to a mixed radical, the index, the denominator and the constant can all be simplified so we simplify.

Week 7 in Precalc 11 – Radical Equations Factoring

This week in Precalculus, we learned how to factor equations with radicals. It’s an important skill since radicals pop up in different units, and knowing how to factor them helps in solving a variety of problems.

Example 1:

step 1: isolate the radical 

 

 

step 2: get rid of the radicand by squaring everything. This removes the radical sign and squares the other side. Be careful because if we have a binomial on the other side we need to write it out twice and foil.

 

 

step 3: move everything to one side so we have 0 on one side. If your squared variable is positive on one side it’s better to move everything there because negative squared variables don’t look nice.

 

 

step 4: factor by prefered method (mine is grouping)

 

 

 

step 5: find out which number makes the equation equal to zero. we might have 1 answer or 2 answers.

Week 6 in Precalc 11 – Quadratic equations and factoring

This week in Precalc 11 we covered the basics of factoring and we also got to touch on solving quadratic equations using factoring.

What makes something a quadratic equation:

  • Quadratic equations are polynomial equations of degree 2.
  • General form: ax^2+bx+c=0 
  • have to make one side equal to zero when factoring
  • there will be up two answers when factoring them
  • follows the zero product law: the zero product law states that if the product of two factors is zero, then at least one of the factors must be zero.

Example 1:

 step 1: Factor – you want to first look for any common variables or numbers, if there aren’t any you can factor by your prefered method (boxing, grouping, inspection, etc) the numbers that are on the other side of the zero. (during factoring quadratic quations we do not touch the zero it just stays there, because essentially the whole point is to get the none zero side to equal zero.)

step 2 : apply the zero product law – applying the zero-product property helps us find the values of that make each factor equal to zero, thus giving us the solutions to the quadratic equation. For example -8 x 8 = 0

 

 

A great way to check to see if you have done your work correctly, is to just plug back in the number to the original equation.

Example 2 : As we can see in this example, we didn’t have all the number on one side and zero on another, but thats okay because all we need to do is to move them to one side. The reason as to why the answers are fractions is because when applying the zero product law, we are taking the equation and isolating x.

 

 

Example 3: in this example we are dealing with fractions. After moving the constant to the left hand side, we multiply everything by 6 because 6 is the least common multiple. then we factor, group and apply the zero product law.

Week 5 in Precalc – factoring polynomials by grouping

This week in Precalc 11 I learned how to factor polynomials by grouping them. This is a good skill to have as polynomials can look scary sometimes and factoring them isn’t easy.

Example: x^{2}+2x+6x+12

Step 1: Start by drawing a square and dividing it into 4 equal boxes inside. Then, place numbers in each box. It’s crucial to arrange these numbers in a specific order so that when you examine them diagonally, the products of the diagonals are equal. Additionally, ensure that similar variables are positioned diagonally opposite each other. For example, if you have 2x in one box, place 6x in the box diagonally opposite to it.

Step 2: Take out the greatest common factor (GCF) or common variable from each row.  This process will result in having two numbers at the top of the box and two numbers at the left of the box. If there’s no apparent GCF or common variable in a row, simply write 1, as 1 is divisible by everything. This step helps simplify the expressions within each row, making further calculations easier.

 

Step 3: Next, let’s group the numbers together. The two expressions at the top of the box should be grouped together, as well as the two expressions on the left. Ultimately, you should have two binomials formed, where the product of these binomials equals the polynomial we began with.

 

 

(x+2)(x+3)

 

Week 4 in Precalc 11 – Rationalizing the denominator

This week in precalculus 11 I was introduced to diving radicals, with this comes rationalizing the denominator. Rationalizing the denominator means eliminating irrational radicals from the denominator of a fraction, thereby converting it into an expression with only rational numbers (fractions or integers) in the denominator.  This makes diving radicals way easier. To start we need to know what a conjugate is. Conjugate is changing the sign of a variable and conjugates are used to rationalize the denominator. 

Example:   if we have 234+47 the conjugate of this number is 234-47 (just changing the sign)

To rationalize the denominator, you need to multiply both the numerator and the denominator by the conjugate of the denominator.

Another example

we know that the conjugate of  sqrt(2) – sqrt(3) is sqrt(2) + sqrt(3), so we multiplied the fraction by that. then we simplified the expression more by 2-3 and then we realized we have a negative denominator which is not good so we multiply everything by -1 so the negative denominator cancels out and we have a nicer number.

Always remember that you need to rationalize the denominator if the denominator is an irrational number

Week 3 in Precalc 11 – Subtracting and adding radicals

This week in pre-calc 11 I learned how to add and subtract radicals. I think adding and subtracting radicals is an important skill to have because it makes working with radicals easy at the end of the equation. To start, we need to know that radicals are treated the same as variables in this lesson, now if we have 2x + 3y + 3x  we get 5x + 3y, this is because only the x’s can be added together since they are the same. to start easily, since we know how to add variables, we can get into adding radicals.

Like variables, radicals that have the SAME radicand go together. here is an example.

 

Now as you can see we only added the coefficients together, you do not touch the radicand, that is just there for you to know which numbers go together.

The same case goes for subtracting radicals. Numbers with the same radicand go together. Here is an example.

However, if we have something like √18 – √8  the roots are the same (both are square roots), but the radicands are different. We’ll start by simplifying each radical. √18 can be simplified to √(9 * 2), and since 9 is a perfect square, it simplifies to 3. So, √18 becomes 3√2. Similarly, √8 simplifies to √(4 * 2), which is 2√2. Now, we have 3√2 – 2√2. Since they have the same root and radicand, we can simply subtract the coefficients: (3 – 2)√2 = √2. So, √18 – √8 simplifies to √2.

 

 

Week 2 in Precalc 11 – Exponents and Radicals

This week in Pre calc 11 I learned how exponents are radicals in disguise. I was familiar with exponents and exponent laws from past math years, but when exploring radicals this year, I learned how exponents and radicals have so much in common. This is an important skill because it was difficult to imagine radicals and work with them in my head but now that I have exponents to help me, it has become way easier to work with radicals. to understand this topic, it is important to remember the exponent laws:

Now that we have recalled the exponent laws we can get into radicals. If we have an example like 4^1/2, we can recognize that the fraction is really 0.5.

what will happen if we put the calculation in the calculator

now isnt 2 the square root of 4? so with this we can conclude that 2 which is the denominator is the root, and 1 which is the numerator is the power of that number

With the knowledge that we have right now with exponents and radicals, we can apply exponent laws ro radical problems and solve them without the need of a calculator.  Remember that the even though we are working with radicals the SAME exponent laws apply  to the problem.

Week 1 Precalc 11 – Number types

During this week in Precalc 11, I learned more about number types. I was familiar with the different number types there are as we have talked about them in Math 10 but I was never confident. I understood all of them this week and can now successfully and confidently identify them.

To start, we need to identify that there are 6 different number types, now there might be more but this is what we are working with in this unit. These 6 different number types are Real, Irrational, Rational, Integers, Whole, and Natural. This order I put them is in descending order. We also need to remember that these numbers have symbols that represent them. Let’s start with the easiest…

Natural  

  • Natural numbers are numbers that our parents taught us. They are the easiest and most simple to work with.
  • In natural numbers there is no 0 only positive whole numbers. 
  • example: 1,2,3,4,5……
  • Symbol: N

Whole 

  • Whole numbers are similar to natural numbers BUT they include zero. 
  • example: 0,1,2,3,4,5…..
  • Symbol: W

Integers 

  • Integers are a little more different, they are whole numbers but they include the negatives. 
  • example: ….-3,-2, -1, 0, 1, 2, 3…..
  • Symbol: I

Rational 

  • rational numbers are where it gets a little more complicated (at least for me).
  • rational numbers can be made by dividing an integer by an integer. like so 👉 a/b    a and b = integers  but b does not += 0
  • rational numbers are repeating like 0.36666666…..
  • but also rational numbers end at some point like 1.3475
  • It is good to remember that rational numbers are ugly numbers that are created by dividing.
  • symbol: Q

Irrational 

  • irrational numbers are different from rational
  • irrational numbers do not repeat at all so they are unpredictable 
  • they also do not end
  • example: Pi, 0.387539821…
  • Symbol: ō (Q with a line on top)

Real 

  •  real numbers are the biggest category and that’s because they include everything
  • so if something is natural it is also real. or if something is irrational it is also real.
  • Symbol: R

You have to remember that a number can be more than one type. for example, 0 is W, I and also R.  but also remember that irrational numbers are very specific and they do not fit this statement. for example, -789 is N, W,I and R but it cannot be irrational.

Here is a drawing to help visualize