I decided to do this because you can skip some steps if you find out the discriminant early ex. if discriminant is negative.

In a quadratic equation, we can determine the nature of x with the help of the discriminant. The discriminant refers to the radical in the quadratic formula,

The nature of x can be determined with the following guidelines.

• If the radical is positive, there are 2 solutions (real & unequal).
• If the radical is 0, there is 1 solution (real & equal).
• If the radical is negative, there is no solution (non-real).

Let’s predict how many solutions a quadratic equation will have and verify.

By this we can see that there will be 1 solution.

Example 2:

As we can see the radical is negative, meaning there’s no real solution. There isn’t really a point in trying to verify since the radical won’t magically change.

Example 3:

The radical is positive, there will be 2 solutions.

If the discriminant isn’t a perfect square, it is usually best to leave the radical as an absolute value. You will still have 2 answers hence the +/- sign.

I chose this because it seems like a useful and easy method to do on most factoring questions that have multiple binomials.

When I’m talking about substituting, I’m talking about replacing a binomial with a variable, for example (3a-4)(a+2) turns into AB, where A is (3a-4) and B is (a+2).

Let’s look at how we can use this to factor.

When we look at the binomials, we notice that we can substitute the binomials for variables to be able to simplify. Let (3a-4) = A, and (a+2) = B

Now the expression has been simplified to

With the expression like this, it is easier to see how we can write this as a product of binomials,

We’re not done yet though, because we have to remember that the variables were substituted. Substituting back gives us:

We can distribute to get ,

Which simplifies to

I chose this because its important to understand the rules to be able to complete the operations correctly.

Dividing monomials is pretty simple, as you simply divide as you think you would divide, so in this post I will focus on dividing binomials, which includes using a conjugate.

What is a conjugate? The conjugate of 1+2 is 1-2, so it’s simply just switching the sign in the middle of an operation.

First, we need the conjugate of the denominator to cancel it out, as we can see the conjugate of is

Next, we multiple the nominator and denominator by the conjugate to cancel out the denominator.

So:

Now we need to properly distribute by multiplying as usual.

is also the same as

This can still be further simplified by multiplying top and bottom by -1, which will reverse the signs.

After that, we still need to distribute in the nominator, giving us:

is also the same as which is the same as

I chose to explain this topic because to me it is more complicated than multiplying radicals, so I think there will be more educational value for me to explain this.

The operations for adding and subtracting radicals are done with mixed radicals, so being able to convert from entire to mixed (and sometimes vise-versa) is crucial.

To be able to combine (through adding/subtracting) radicals, the index and radicand have to be the same.

As said before, we have to convert to mixed, which is done the same as before.

In mixed radical form:

After converting to mixed, we can see that the radicals have the same index and radicand as some other radical, so now we actually add and subtract. To do this we just combine the coefficients with like terms (remember, like terms in radicals: same index & radicand)

So, 3+2 is 5, and -4-2 is -6, meaning the answer is

In the case of operations on unsimplified mixed radicals, the mixed radical still has to be in simplest mixed radical form.

While these radicals are mixed, they still can be further simplified.

63 becomes root 7 with a coefficient of 3, but we can’t just ignore the other coefficient. to combine them, simply just multiply the coefficients. multiplied by is 1.

Do the same thing to the second radical, and you will find out that they have the same radicand in simplest forms, meaning you can further simplify.