This week in Pre-Calculus 11 we learned about discriminant’s and how they can be useful. A discriminant is a portion of the quadratic formula that you can use to predict how many solutions you will have, based on if the value is positive, negative, or equal to zero. When graphing the discriminant represents how many times that the parabola touches the x axis. You should also note that finding the discriminant is NOT a way of solving quadratic equations.

Note, if the value of the discriminant is:

• Positive: then that means that you will have to solutions
• Negative: then that means that there are no solutions
• Equal to zero: then that means that there is one solution

The formula for finding the discriminant:

*highlighted in yellow*

Example:

Touches the x axis in two spots

1. Firstly, since our equation is already written in the proper format of (), we need to write down the values for a, b, and c. (a = 2, b = 9, c = -4)
2. Next, we input our values into the formula.
3. Use BEDMAS to simplify the equation until you are left with a value.
4. Since the number we are left with is 113, and it is positive, then we know that this quadratic equation will have two possible solutions.

Example:

Doesn’t touch the x axis

1. First we need to identify the values of a, b, and c. (a = 6, b = -2, c = 7)
2. Next we input the values that we know into the formula.
3. Use BEDMAS to simplify to a value.
4. Since the value that we are left with is 164, and it is negative, we know that that this quadratic equation will have no solutions.

Example:

Touches the x axis in one spot

1. Firstly, since our equation is not in the proper format we need to rearrange it so that it is.
2. Now that the equation is in the proper format we can write down the values for a, b, and c. (a = 1, b = -8, c = 16)
3. Input the values that you know into the formula.
4. Use BEDMAS to simplify to a value.
5. Since what we are left with is zero, we know that this quadratic formula will have one solution.

This week in Pre-calculus 11 we learned a number of different ways to solve quadratic equations. One method that I liked was the quadratic formula. The quadratic formula can be used to solve any equation (rational, irrational, etc.), even ones involving fractions!

The quadratic formula is based off of the values in a general quadratic equation:

• 
• In this equation a is not equal to zero, a≠0

The quadratic formula:

Example:

1. Since the equation is set up in the pattern that we need it in we can find the values of a, b, and c. (a=2, b=-10, c=12)
2. since we have the values for a, b and c we can input them into the formula. (the two negatives in front of the 10 cancel each other out and make it a positive). 
3. After writing in the values you need to use BEDMAS to simplify the radicand in the numerator.
4. You then need to check if you can simplify the radical. We know that four is a perfect square, so we can write it as a whole number ().
5. after simplifying the radical you check to see if any of the coefficients have anything in common so that we can simplify the fraction. In this case all of the coefficients are divisible by 2.
6. Since our answer can’t be further simplified we leave it as is.

Example:

1. Since our equation isn’t written in the proper format we need to rearrange it. We do this by moving the 5 over to the left side.
2. Now that it is written in the proper format we can find the values of a, b and c. (a=1, b=3, c=-5).
3. Next you need to input what you know into the formula.
4. Simplify the radical using BEDMAS.
5.  Check to see if the radical can be simplified. can’t be written as a mixed radical.
6. Check if the coefficients have a common factor. (-3, 1, and 2 have nothing in common).
7. Since we can’t further simplify the answer we just leave it as is.

This week in Pre-calculus 11 we learned about radical expressions and how you would solve them. We learned that in order for you to solve a radical expression you must understand that square roots and squaring are inverse operations. Inverse operations are opposite operations. ex. addition is the opposite of subtraction.

Before solving a radical expression, you must recall:

• What you do to one side you must do to the other
• You need to isolate in order to get a variable by itself
•  When you divide by a negative number in a inequality you have to flip the sign (this is only important when you are writing your restrictions)

Note: when writing your restrictions you must write whatever is under the square root sign and isolate to find the restrictions for x.

ex.   = 2

Restriction:

• 2x+6 ≥ 0
• 2x ≥ -6
• x ≥ -3

Example:

1. First, we need to get rid of the sign. In order to do this we put brackets around it and square it. Since we square the left side we need to square the right side as well.
2. In the expression x is already isolated we don’t need to move anything around; x = 36.
3. Since all expressions involve variables, after finding a value for x we need to write a restriction. Since it is a square root x needs to be greater than or equal to zero, x  ≥ 0. We then double check that our value for x fits into our restriction.
4. After solving for x and making a restriction you then need to do a check to make sure that you got the correct value for x. All you do is input your value for x into the expression and solve it. Both sides need to equal the same number, if not, you need to go back and double check that you solved the expression correctly.

The steps for solving an expression are:

1. Solve for x
2. Write a restriction
3. Check

Example:

1. Since there is a coefficient of 4 in front of the ,we divide both the left and right side by 4. By doing this it makes the expression easier for us to solve. .
2. We then square both sides to get rid of the .
3. Since we have found the value of x, we need to write a restriction. Since it is a square root x needs to be greater than or equal to zero, x  ≥ 0. Check that your value for x fits into the restriction.
4. Check.

Similar Example:

Example:

• Sometimes there will be instances where there is No Solution to the expression. These expressions are referred to as Extraneous Solutions. You will be able to tell if an expression is an extraneous solution if the square root in a question is equal to a negative number. The reason that this type of expression will have no solution is because it isn’t a true root. No square root should be equal to a negative number just like how you can’t have a negative number as the radicand.
• If you come across an expression where the square root is equal to a negative number, simply write no solution.