This week in math we learned about a specific part of the quadratic formula called the discriminant that can help you determine the solution of a quadratic equations faster. I choose this because it sounds like a useful tool to use on the mid term. it is important because in can save us the work of having to go through the entire quadratic formula if there isn’t any solutions and allows us to use other much faster ways of solving such as inspection and the area model while knowing how many solutions there will be and wether or not they will be rational or irrational.

1. Sure, I can explain how to use the discriminant formula step by step. The discriminant is a part of the quadratic formula used to determine the nature of the solutions of a quadratic equation. Here’s how you use it:1. Understand the Quadratic Equation: First, make sure your equation is in the form of , where ( a), ( b ), and ( c ) are constants, and ( x ) is the variable.2. Identify the Coefficients: Identify the values of ( a ), ( b ), and ( c ) in your quadratic equation. For example, if your equation is , then a = 2 , b = 5, and  c = -3.3. Calculate the Discriminant: The discriminant (\( \Delta \)) is calculated using the formula . Substitute the values of a, b , and c  into this formula and perform the calculation. Continuing with the example, .4. Analyze the Discriminant:
   – If , the quadratic equation has two distinct real solutions.
   – If , the quadratic equation has one real solution (a repeated root).
   – If , the quadratic equation has no real solutions (the solutions are complex).5. Find the Solutions (If possible):
   – If , use the quadratic formula to find the two distinct real solutions.
   – If , the repeated root is given by .
   – If , there are no real solutions.

Example:

let’s solve the quadratic equation \( 8x^2 – 10x + 3 = 0 \) using the discriminant formula.

1. Identify the Coefficients:
– \( a = 8 \)
– \( b = -10 \)
– \( c = 3 \)

2. Calculate the Discriminant:
–
–
–
–

3. Analyze the Discriminant:
– is greater than 0, which means the quadratic equation has two distinct real solutions.

4. Find the Solutions:
– Use the quadratic formula to find the two solutions:
–
–

So, the solutions to the equation are and .

Would you like to try another example or explore anything else related to quadratic equations?

The week in math we learned about at new way to solve quadratic equation using the quadratic formula which is a formula that uses the a quadratic equation in standard form and is important inorder to quickly determine wether a quadratic equation is solveable and can be in some situations a faster method in solving than the box method inspection or perfect square. I choose this topic because my physics teacher talked about using it in my kinematics units and it sounded interesting.

To use the quadratic equation , follow these steps:

1. Identify the coefficients: Determine the values of in the quadratic equation.

2. Plug into the quadratic formula: Substitute the coefficients into the quadratic formula:

This formula gives you the solutions (roots) for the quadratic equation.

3. Calculate the discriminant: Compute , which determines the nature of the roots:
– If , there are two distinct real roots.
– If , there is one real root.
– If , there are two non-real roots.

link

Example:
Suppose we have the quadratic equation we dont have to change anyting since the coefficient for the high power is positive and all the terms are on one side .

1. Identify .
2. Plug into the quadratic formula:

Simplify to get:



So, the solutions are and .

3. Calculate the discriminant:

Since the discriminant is positive, there are two distinct real roots.

Therefore, the solutions to the quadratic equation are and .

I chose solving Quadratic Equations because they were an interesting application of factoring and grouping and past learning of solving for x. solving Quadratic Equations is important because they allow us to see where linear and non linear lines intersect and define variables in more complex equations.

In order to solve a quadratic equation we must identify if an equation is quadratic by seeing if there is a power of 2 and if there is an equals sign. first Write Down the Quadratic Equation:Start with the quadratic equation in standard form: ax^2 + bx + c = 0. where the all the terms are on the side where the variable with an exponent of 2 is positive. Next try to factor the quadratic expression into two binomial factors. This step involves finding two numbers that multiply to give you the constant term (c) and add up to give you the coefficient of the linear term (b) unless the highest power variable has a coefficient higher than one we would do grouping. Once you have factored the quadratic equation, set each factor equal to zero. This creates two separate linear equations. Solve each of the resulting linear equations for the variable x. Check the solutions you found by substituting them back into the original quadratic equation. If they both sides of the equation are equal, they are the correct solutions. Show the solutions in simplest form.

1. Write Down the Quadratic Equation: x^2 – 5x + 6 = 0
2. Factor the Quadratic Equation: since the coefficient of the highest power variable is one we need to find two numbers that multiply to give 6 and add up to give -5. These numbers are -2 and -3. So, we can rewrite the equation as: (x – 2)(x – 3) = 0.
3. Set each factor equal to zero: x – 2 = 0 and x – 3 = 0
4. Solve each linear equation: Solving these equations gives us: x = 2 and x = 3.
5. Check Solutions: Substitute x = 2 and x = 3 back into the original equation: (2)^2 – 5(2) + 6 = 4 – 10 + 6 = 0, and (3)^2 – 5(3) + 6 = 9 – 15 + 6 = 0. Both solutions satisfy the equation.
6. Write Down the Solutions: The solutions are x = 2 and x = 3.link

I choose the grouping box method to factoring polynomiols because it was a good way to visualize factoring and to show how the gcf applied to it in a way i hadn’t noticed before. this is an important foundational tool in order to understand how the terms in a polynomial work before using more efficient methods that require this knowledge. in order to factor polynomials using this method we must have 4 then we place them in 2×2 box and then find the gcf of each column and row pair effectively grouping them. after that wew up  place the gcf  on the outside of their columm or row on the same side and then we find are factors by grouping the terms on the vertical and horizontal sides of the 2×2 box here is an example.

1. in order to factor with the box method we must make sure we have 4 terms like this polynomial ” +2x+3x+6″

2. then we must first make a 2×2 box

3. put the terms in a correct order where the values with a variable at a power of 1 which are 2x and 3x goes in the top right and bottom left

4. put the constant or the term without a variable in the bottom left and the last term with a variable that has a power of 2 in the remaining box like this:

link

5.  find the gcf of both collums and place them at the top of their  column which would be x for (, 3x) and 2 for(2x, 6) meaning that one of our factors is x+2.

6. find the gcf of both rows and place them to the left of their row which would be x for(, 2x) and 3 for (3x, 6) meaning that one of our factors is x+3.

7.  put our factor together to find the factored form of this polynomial which is (x+2)(x+3) and the box with the factored form looks like this:

I choose dividing Radicals because it is has a unique aspect of it which is rationalizing the denominator. it is important inorder to simplify more complex radical expressions. this can be done by simply dividing the coefficients and radicands by each other but sometimes we are asked to rationalize the denomitor this is a unique way to simplify radicals when dividing and happens when we have a denominator that is a radical we must simplfy it which means not only puting the entire radicals as mixed but also rationalizing the denominator which involves getting rid of radicals from the denominator of a fraction. Here are the steps to do it:

1. Identify the Radical: Identify the square root (or other root) in the denominator of the fraction.
2. Multiply by the Conjugate: Create a new fraction with the same numerator, but multiply both the numerator and the denominator by the denimator or the conjugate of the denominator expression. we use the conjugate when the dinominator is an expression with atleast two terms which is called a binomial  and is obtained by changing the sign between the terms of the radical for example the conjugate for this binomial “” is this ““.
3. Apply FOIL or Distribution: Multiply the numerator and the denominator using the distributive property (or FOIL if there are two terms) to simplify the expression which just means multiply the first term against the both terms of the other binomial and the same with the second.https://study.com/learn/lesson/foil-method-math-overview-examples.html
4. Simplify: After multiplying, simplify the resulting expression as much as possible by combining like terms such as the indexs, coefficients and radicands.
5. Check for Radicals: Ensure that there are no radicals remaining in the denominator. If there are, repeat the process until the denominator is free of radicals.

By following these steps, you can rationalize the denominator of a fraction effectively.

Example:

This example has us, use already simplfyied terms so we only have to rationalize the denominator which we do by multiplying both sides by the conjugate of which is this shows us a difference of squares where the 3rd term so that gets cancelled out we then get 1 from 4-3 since =4-3 then we multiply 5 by the conjugate so that the value of the term doesnt change like multipling it by one we then distribute 5 to so we get which is are answer in simplest form

I choose Adding and subtracting radicals because it was a new to use radicals in a brand new way to uses my past math knowldge like keeping the same base with exponents or denominator with adding and subtracting these operations. Adding and Subtracting radicals is important Step in order to simplify radical equations to there simplest form

1. to add and subtract radicals we must first make sure that the radical are like which means that the radicands or numbers in the root sign and the index of the radicals used are the same then we simply leave the radicand and add or subtract the coefficients for the result
2. an example of this is this to solve we add the 3 and 4 coefficients to get 7 and leave the radical 4 this is the answer:  .
3. if not we must first simplify one number to its  simplest form in mixed radical form this can be done by finding out two numbers that multiply to equal to it with one        being a perfect number then we simply  put that numbers root as the coefficient and the other as the the radicand furthermore if you put both in simplest mixed fraction form and they dont have the same index and radicand then leave them in mixed fraction form.
4. after the bases are the same we can add simply add or subtract the coefficients and leave the radicand .
5. another example is this this example doesn’t have like radicands so we must simplify the big radicand 27 to do this we first find the perfect square since the index or root is 2  which we know is 9 then we know that 3 is the square root of 9 so that is the coefficient and 9 times 3 is 27 so the radicand is 3 which equals then we know that
6. picture link: https://www.youtube.com/watch?v=qkwjfwtE6uc&ab_channel=MichelvanBiezen

I chose to use a fractional exponent because it is a powerful mathematical tool that allows us to represent roots and rational powers in a concise and efficient manner and show numbers that cant be represent in integer form such as irrational and undefeined numbers. Fractional exponents are a way of expressing powers and roots that are not whole numbers, essentially serving as shorthand form for roots and powers of numbers.

firstly fractional exponents are a way of using normal powers to represent the radical for example when we have an expression of the form ​, where “$a”$ is a base, $m$ is the numerator or the top number, and $n$ is the denominator or the bottom numbers, this represents the $n$th root of “$a”$ raised to the power of “$m”\; which\; is\; how\; many\; times\; you\; multiply\; itself\; onto\; itself\; with\; an\; index\; or\; root\; of\; “n”\; which\; is\; how\; many\; of\; the\; same\; number\; multipylied\; by\; each\; other\; is\; needed\; to\; equal\; this\; one$.

fractional radical also let you apply the laws of exponents, allowing for simplification of expressions and equations involving roots and powers allowing you to reduce the amount of steps required to use radicals in equation. for example we can multiply two numbers together and instead of converting them to the same form if one is an integer and the other is in fractional exponent form we then simply combine there powers together as shown in the below picture of product of powers and the fractional powers law an example would be instead of “” which takes other steps and is harder to solve we can translate it to this and use the product law which means we add the exponents when the bases are the same in multuplication “” which we know equals to or 64

furthermore, fractional exponents can also be shown as a negative powers allowing for more equation we can use radicals in and they can also be converted to radical form if required, further helping in simplification for example is equal to . Additionaly fractional exponents can represent irrational roots, providing a important tool in mathematical problem-solving to use them in a wide variety of way. For instance, ​ represents the cube root of 9, which is approximately 2.08, an irrational number. In summary, fractional exponents offer a simple and efficient way for expressing roots and rational powers of numbers, with a advandced understanding of their values significantly simplifying mathematical expressions and equations.

Exponent laws

photos: https://www.easysevens.com/laws-of-expeonents-and-logarithms/

This week, we explored the principal square root, signified by the symbol “√”, which stands for the positive square root of a number. Knowing this concept is vital for answering questions, aiding in answer determination. When a question doesn’t include the square root symbol, it means you should consider both the positive and negative square roots in your response represented by the “±” symb0l. I selected this concept because it serves as a solid foundation for understanding various square roots of numbers.