As the final blog post of Math-11, I am doing a top 5 list of things that I learned this year.

1. Solving with quadratic formula
2. Sine & Cosine law
3. Solving by completing square (& vertex form)
4. Discriminant
5. Entire radicals to mixed

Explanation

The quadratic formula​​ provides a straightforward method for solving any quadratic equation regardless of whether the roots are real or complex. Unlike factoring or completing the square, which may not always be easy or possible, the quadratic formula works in all cases where a, b, and c are known.

The Sine and Cosine Laws are essential for solving non-right triangles, which frequently occur in real-world problems, such as engineering, navigation, and physics.

Completing the square is (usually) a fast way to solve/simplify and it provides insight into the visual representation of quadratic equations (vertex form).

The discriminant tells us the nature of the roots of a quadratic equation (real and unequal, real and equal, or complex/non-real).

Converting entire radicals to mixed radicals simplifies expressions and makes them easier to work with, especially in algebraic operations.

I chose to talk about these laws because it will definitely be important for grade 12, as they allow to solve non-right triangles.

The Sine Law relates the ratios of the lengths of the sides of a triangle to the sines of its angles.

The Cosine Law relates the lengths of the sides of a triangle to the cosine of one of its angles.

Formulas:

Sine Law:

Cosine Law:

Examples:

Given A=30°, B=45°, and a=10:

Given a=7, b=10, C=60°

As you can see, use sine when you know two angles and one side or two sides and a non-included angle. Use cosine when you know two sides and the included angle or all three sides.

I chose to do this because it is one of the more challenging things to do, so explaining it will help me learn more.

Algebraic fractions basically just means the fractions have variables. The rules don’t exactly change, but more are added on (exponent laws, factoring, simplifying, restrictions).

There are a few more things to watch out for, like opposites always cancelling out to -1 and binomials sometimes being the LCMs, though same rules apply.

You can see that (x+5) is the LCM, so it will be cancelled out anyway, meaning we can ignore it and write it once in our denominator.

To make a common denominator, we will need to multiply both sides by what will give them the common denominator, which is (x-3) on the left and (x-2) on the right

Now we can subtract

Note the minus sign in the middle, which will flip the signs of the fraction to the right.

In this fraction, we can see the the denominators can be factored.

Again, we have an LCM of (x+1)

Last week I talked about how to graph linear inequalities, so I think it’s fitting to talk about how to graph quadratic inequalities this time.

Like last time, the first thing we do is graph it as an equation.

Turn into standard form, we can treat it as a quadratic equation for now.

Now, to make this an inequality, we basically do the same thing as with linear equations.

Choose a point, and if the statement is true then that’s the side of the line that needs to be shaded. Concerning the line itself, the rules are also the same. Dotted line = not including.

0<(0-4)^2-6

0<16-6

0<10 is true

To add on a little more, we can write this in interval notation.

We start by factoring, but quickly looking at the original quadratic we can tell it’s not factorable, which isn’t that big of a deal as we have alternate methods such as using the quadratic formula.

Now we have the x-intercepts, and we can determine where the quadratic is above the x axis, since we need to know where the quadratic function is greater than y=0.

The parabola opens upwards in both directions to infinity, so the parabola is greater than y=0 in both of those directions.

I chose to do this because it’s new and will probably be important for later.

To do a quick review of inequalities, they are expressions that compare 2 values. It’s like an equation, but the equal sign is replaced with different symbols.

Solving inequalities is like solving equations, but graphing is a bit different.

Specifically, we are going to be looking at the inequalities with 2 variables, giving us a x and y axis.

To make our initial line, it’s pretty simple. Start as you do usually, I like setting x to 0 to find the y intercept and start graphing from there.

From here, we can do the classic rise over run. Since x has a coefficient of 5 (5/1), we go up 5, and right 1.

The graph will look like this.

Obviously, so far we just graphed an equation. There’s no inequality.

To be able to graph inequalities we need to understand a few things.

1. The type of line. if the sign is a regular greater than/less than sign, the line will be dotted. If the sign is a greater than or equal to/ less than or equal to sign, the line will be filled. This is to show whether or not the line itself is considered a solution.
2. Filling the right side of the graph. We need to distinct the solutions from the non-solutions. We do this by coloring/filling the area that has solutions. We can do a test on the inequality. Choose a point, like (0,0) to put into the inequality. If the inequality stays true after having input the point, then the side of the line with that point will have the solutions.

As we can see, this inequality is not true, so the solutions will be on the other side of the line.

To sum it up, the graph will have a dotted line, and the shaded area will be on the other side of the line relative to (0,0).

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.