Follow along through my example and you will hopefully be able to see what I mean.

Let’s consider the equation x² + 7x + 10 = 0

First, we can identify that there is an x² term (with a non-zero coefficient… 1), so we can say that it is a quadratic equation.

To solve a quadratic equation, we want to determine the roots, or what values make the equation true. To help us to achieve this, we want to rearrange the left side so that it is a product of two terms (or expressions). In this way, we can say that “something times something equals zero”. And since we need one of those “somethings” to be zero if the product is zero, we essentially break this down to “something #1 = 0” and “something #2 = 0”, and by solving these two simpler equations, we will arrive at our roots. So, continuing with our example then, let’s factor it. Review my post on methods of factoring if you need a bit of a refresher!

x2 + 7x + 10 = 0
(x + 2)(x + 5) = 0

This is what we’re looking for: two expressions multiplied together to give zero. Now, we have two equations to work with to find our roots of the quadratic equation. Rewriting, this gives us:

x + 2 = 0 and x + 5 = 0

And quite obviously, these can be solved to show that x = (–2) and (–5). And since we followed that whole process, we can consider these two values to be roots of our original quadratic equation. However, it is VERY IMPORTANT to substitute these values back into the original equation to check! With these values, we can show that:

(–2)2 + 7(–2) + 10 = 0
4 – 14 + 10 = 0…….. this is true. So –2 is for sure one of the roots. I’ll leave –5 for you to verify on your own.

If you find a question and proceed all the way through to find the roots, and you go and plug them back into the original equation, if one of the roots does NOT satisfy the equation, you cannot count it as one of the roots. This sometimes happens when you have an expression in a denominator (eg. (x – 2)), and if you determine through the above steps that your expression gives you a root of 2, by plugging this into your original equation, specifically into the denominator, the denominator will equal 0 and cause the expression to be undefined. Therefore, this root does not satisfy the original equation and you just ignore it.

I hope this has helped to explain the process you need to follow to solve quadratic equations. With practice, they will become second nature. However, despite all of the work required, sometimes it just is not practical or apparent how to factor your quadratic equation. In these cases, you would likely want to rely on the use of the quadratic formula, which I will go over in a future post to explain what it is and how it works. Let me know if this makes sense or if you’d like anything more added.

When most students hear “quadratic equation,” they usually get anxious because quite often this means having to work with the quadratic formula. This formula is more complicated than most that you have probably encountered up to this point, but factoring quadratics doesn’t always rely on the quadratic formula! In fact, they can be quite simple! A quadratic equation isn’t just “something that needs the quadratic formula” to solve it. Quite simply, a quadratic equation is just an equation that can written in this form:

ax² + bx + c = 0 where a, b, and c are real numbers and a does not equal 0

See? That doesn’t sound so bad. The KEY is that the “a” value is not 0. b and c can be, but not a. You need to have the x² term.

Remember, “solving an equation” means to find the roots or solution… or, what makes the expression true? To do this with quadratic equations, we rely on the property that says “two terms multiplied = 0 only if one or both of those terms is 0.” Remember this property! It is key to the quadratic factoring method. If we combine this property with our ‘grouping’ factoring method, you will see how this all comes together.

For my first example of using the quadratic formula to find the roots of a quadratic equation, let’s keep it simple.

x² – 2x –3 = 0

Comparing this to the standard form of a quadratic equation, ax² + bx + c = 0, we can equate the letter coefficients to the values provided. That is, we can say that a = 1, b = (–2), c = (–3). Now, we can simply substitute these values into the quadratic formula:

So, we have:

If you follow along with the arithmetic, you can see that we’ve solved the quadratic formula to show that the roots of the given equation are x = 3 and x = (–1).

Now, remember that I said in a previous lesson that you have to check your answers! Substitute these values back into the original equation, and you will find that they do indeed satisfy the equation. So, these are the correct roots!

Of course, you may have noticed that this question didn’t actually require the quadratic formula to solve for the roots. The quadratic formula worked well and got us the answer, but as you saw, it required a bit of work. And more work means more opportunity to make a mistake! You may have noticed that there was actually a faster way of solving the question. If you noticed that you could reduce the question down to (x – 3)(x + 1) = 0, you could simply let each set of brackets equal zero, and then find again that x = 3 and x = (–1) are the correct solutions.

Let’s try another one, adding some more of the previous math concepts I’ve gone over.

Using the quadratic formula, find the roots of:

2x³ + 3x² = 4x

It’s looks a little more complicated than the last one, huh? It has higher order exponents, and it doesn’t immediately look like a quadratic equation, as the first example did. However, with a little bit of arithmetic, and using your skills from the math concepts I explained in my post about factoring, it will begin to look a bit more familiar and workable.

So then, apply grouping techniques to our question. Let’s bring everything to one side first though. Recall that the standard form of a quadratic equation equals zero.

2x³ + 3x² = 4x
2x³ + 3x² – 4x = 0
x(2x² + 3x – 4) = 0

Looks a little better now, right? Maybe, something that might fit into the quadratic formula? Recall that the roots, or solutions, are any values of x that make the expression true. So, what we have derived up to this point is a product of two expressions that equals zero, and therefore the roots will be whatever values of x cause each part of the product to equal zero. The first (potential) root is obvious, from the first of the two expressions in the product: x = 0. (Substitute 0 back into the original equation to verify this is a correct root!) The second part, 2x² + 3x – 4, will require more work, and if we let it equal zero, you can see that it will fit into the quadratic formula perfectly.

To prepare for the quadratic formula, we need to identify our a, b, and c values. They are: a = 2, b = 3, and c = (–4). Now, we just substitute into the formula, do the math, and come up with our root(s) for this part of the question!

So, these are our answers for the two roots to the quadratic expression part of our original question. These are the radical forms of the solutions, so they look way more complicated. But, often the quadratic formula doesn’t reduce all the way down to a nice, round number and you will be left with something like this. The last thing you have to do is substitute them back into the original question to verify the roots are true, and that is it! Of course, when you write your answers down, make sure you remember to include the roots from the first part of the question, i.e. the part we created by grouping and solved for x = 0.

That last question goes over a lot of math concepts and is definitely comparable to some of the more complicated math questions you may find in your homework or on exams. Review and study it and make sure you understand it. I’ll post another example as well soon, if anyone needs some more examples of using the quadratic formula.

However, as I said, not all quadratic equations can be solved this way. Sometimes, they are already expressed in a simplest form, or further manipulations just make things messier. In these cases, you can use The Quadratic Formula to solve for the roots of the equations. At first glace, the quadratic formula looks like a beast of a formula to use, and even harder to memorize! But, trust me… commit this formula to memory and learn how to use it, and solving quadratic equations will become so easy for you!

So, what is the Quadratic Formula?

I will go over how to solve it, but first, the it looks like this:

You can use this for any quadratic expression of the form ax² + bx + c = 0, where “a” does not equal zero. (If you think about this condition, you can see that if a = 0, then there is no x² term at all, and you are left with a linear equation or something of a higher order. Also, if a = 0, the quadratic formula then has 2(0) in the denominator, which equals 0 and causes the whole expression to be undefined. So, hopefully that short explanation will help you to remember that if a = 0, you cannot use the quadratic formula!)

Working through the math of the quadratic formula isn’t as difficult as you may think. To start, all you do is arrange your question into the form of ax² + bx + c = 0, and then you can easily identify the coefficients for a, b, and c. Then, you simply substitute those values into the quadratic formula, and do the math. One thing to draw your attention to though is the “plus/minus” sign. Basically, the quadratic formula is really TWO formulas, one with a “-b + √…..” and one with “-b – √…..” These two formulas are what give you your two roots.

You will study this more in the future, but for now you may find it interesting that a quadratic equation, i.e. an equation with an x² term, defines a parabola. The equation of all parabolas have x² as the highest order exponent. As a result, you can imagine that a parabola drawn on an X-Y graph will cross the x-axis twice (at the most). These are the roots or solutions of the equation, and so that is why you cannot have more than 2 roots. Similarly, you can figure out why there may be 1 or even 0 roots, depending on where the parabola is located on the graph.

So, now that you know the answer to ‘what is the quadratic formula,’ next I will show you how to use it. Examples coming in my next post.

To start, I will explain first-degree equations in one variable. Quite simply, a first-degree equation is one in which there is only one variable. In general, a first-degree or linear equation has the form:

ax + b = 0, where a and b are real numbers and a does not equal 0

I’m sure you are extremely familiar with this type of equation, though you may not know it by this name. An example of a linear equation would be something like 3x – 12 = 0. Undoubtedly, you can easily see that this equation is true when x = 4. However, it is good to realize that the expression is neither true nor false until you substitute in a value for the variable. Any value that makes the expression correct is called a solution or root of the equation. To further classify this equation, 3x – 12 = 0 is also called a conditional equation, in that it is only true for certain values.

When you have two expressions that have the same solution (or root), these are called equivalent equations. Again, I’m sure you are familiar with the concept, but probably unfamiliar with this name. When you have a first-degree equation in one variable, the general strategy that you typically employ is to express the equation equal to a series of equivalent equations, which you manipulate until you can reduce everything down to the solution to the equation.

The rules for generating equivalent equations are simple and intuitive:

• You can add or subtract the same value from both sides of the equation. (A corollary to this is that you can add AND subtract the same value on one side, without changing the other… since adding x and then subtracting x means you really have done nothing!)
• You can multiply or divide each side of the equation by the same value.
• You can simplify one side of the equation without affecting the other side of the equation.

I think these rules are fairly self-explanatory, so I’m not going to bother going into any examples to demonstrate them.

When you have arrived at your solution / root of your equation, it is ALWAYS smart to take that value and substitute it into the original expression to verify that it is indeed true. It always amazes me how many people arrive at incorrect answers and leave it at that, when a simple review and check can either tell you that you are correct, or your answer needs more work. ALWAYS REMEMBER TO CHECK YOUR ANSWERS!

By checking your answers by substituting the solution into the equation, you sometimes will determine that the solution you have found CANNOT be true, in which case your solution is called an extraneous root. An example of this would be where, when checking your solution, you determine that you have a 0 on the bottom of a fraction (the denominator). A fraction with a zero in the denominator is undefined, and so you can conclude that the root you determined does not satisfy your equation. Extraneous roots may develop especially if you use rule number 2 above, but you multiply both sides by an EXPRESSION rather than a single number. (eg. you multiply both sides by (x + 2))

That is all I am going to say about how to solve equations for now, especially the first-order equations (or linear equations). I will continue in my next post with a discussion of solving quadratic equations.

First of all, a high level, general definition is needed. What does factoring mean? Factoring means to simplify a mathematic expression by writing the expression as a product of two or more values or expressions. For example, if you factor the expression 12x + 4, you get (4)(3x+1). This is an example of “factoring out” the 4. You will see this phrase “factoring out” very commonly when dealing with these types of problems.

Several factoring techniques are available to you to help you find factors, depending on the question. These are some techniques you can use. You can memorize these factoring tricks and shortcuts and you will save a lot of time with your math work! If you do lots of practice with factoring games or math worksheets, you will become good at these and be able to find factors very quickly.

Common factor

In an expression composed of multiple terms, try to identify if there is one number/variable that is a common factor to each term. Then, after factoring out the common factor, you can rewrite the expression to show multiplying that common factor by the remaining terms.

12x + 4 = (4)(3x + 1)
5x³ + 10x² + 25x = (5x)(x2 + 2x + 5)

Difference of squares

Wherever you see a difference of two terms that are perfect squares (either something like x² or 25), you can apply this technique for factoring a difference of two squares. This actually is the same as one of the rules for special polynomial products. In fact, factoring special products follow the same rules you would use to find factors anywhere else. Factoring squares is actually quite simple:

x² – a² = (x – a)(x + a)

You reduce the terms to their square root value, and remember to put one ‘+’ and one ‘-’. Easy as that.

Difference of cubes / Sum of cubes

Factoring perfect cubes (or factoring cubics) is a little trickier, but they follow a strict form that you can memorize and use easily to find factors. Depending on if you are subtracting perfect cubes or adding perfect cubes, you will use the appropriate formula:

x³ – a³ = (x – a)(x² + ax + a²)
x³ + a³ = (x + a)(x² – ax + a²)

They have the same basic form, you just have to pay attention to the signs. Don’t mix them up!

Grouping

This is an extension of the common factor method described above. The only difference is that the common factor doesn’t have to be common with EVERY term. Group things together, and factor within the groups. Take this example and you should see what I mean:

x³ – x² + x –1
= (x³ – x²) + (x –1)
= x²(x – 1) + (x – 1)
= (x – 1) (x² + 1)

Study that example and it should be fairly self-explanatory how I used grouping to find factors here.

Trial and Error

Sometimes, there aren’t any obvious factoring tricks or factoring techniques that you can apply to help you solve your question. Unfortunately, in these situations, you must resort to trial and error. Sometimes you can figure out the numbers that are involved, but you need to test out the signs to get it right. I can’t really say anything about this technique except to have patience and keep trying.

I will revisit this post shortly to put up a remark about factoring quadratic equations. I will also go over AC method factoring. The AC method of factoring is a factoring method or factoring trick you can use to help you factor expressions in the form of ax² + bx + c (trinomial).

To begin, let’s start with this equation:

Now let’s rearrange a bit by multiplying both sides by r²:

We currently have in this expression a natural logarithm. Since the natural logarithm ln(x) is defined as the inverse function of the exponential function e^x, we can remove the logarithm from our equation entirely by raising the number e to the power of our equation:

Now, since ln(e^x) is equal to x, we can apply this to our equation to leave only what was in the parentheses on the right side:

To remove the denominator, we have to multiply both sides by m:

Finally, with a little bit more rearranging, we are left with our final expression:

Merry Christmas and Happy Holidays to all of my readers, and best wishes for the New Year!

Google has included many special features into their search page that aren’t immediately obvious to the users. For example, you can type in New York weather to get a weather report for New York (instead of getting search results for weather websites). Or, if you search for your favourite sports team, you can get the latest score from their game. Of particular interest to this post, however, is the built-in calculator function.

To access Google’s calculator, all you need to do is type “calculator” or enter the mathematical formula into the search box, and you will be presented a result in an online calculator app. From there, you can make changes to your formula, or clear it and start over again. This calculator even goes beyond basic math functions, by providing buttons for trigonometry, logarithms, exponents, etc. You can even change between Radians and Degrees modes, just like on a real scientific calculator.

This calculator isn’t just limited to being available on desktop browsers. You can access it just the same from mobile browsers on your iPhone or other smartphone. The mobile version has a slight variation to it, in that you are shown a basic calculator when you hold your phone in portrait mode, but flip it horizontally into landscape mode, and it turns into the full-size scientific calculator.

Also, if you have Desktop Voice Search enabled on your Google Chrome browser, all you need to do is click the mic icon, and speak your math equation. Google will interpret your words, and return the calculator to you with the result. Related to this Google calculator is the unit conversion trick you can do in the search box. Simply type in something like “3.25 miles in km” and it will do the conversion for you in the result.

These tricks are great time savers if you find yourself without your calculator, or you just want to get a quick answer to a math question without having to bother going to look for your calculator. And the way website browsers are designed now, it’s takes almost no time to turn on the device and get to a search box, where you can enter your math questions!

Briefly, to summarize David’s work, you want to think of the two numbers you are multiplying together as being two “groups.” After that, the key to multiplication is to understand that it is a form of repeated addition. That is to say, if you multiply 2 x 3, you are really adding 2 + 2 + 2, or 3 groups of 2.

So, if you have 5 x 2, it is really saying “add 2 groups of 5,” or 5 + 5. (Of course, it could also mean add 5 groups of 2… they both mean the same thing.)
Similary, if you have 10 x 4, it is telling you to “add 4 groups of 10,” or 10 + 10 + 10 + 10. (Or similarly, 10 groups of 4.)

Multiplying two positive numbers is easy. But what about if one of the signs is negative? What does that mean? Well, if you look at it as groups, and remember the key of repeated addition, you will hopefully be able to understand it.

Take 2 x (–5). Using our language, it is telling us to add 2 groups of (–5). The sum is now a larger group equal to (–10).

Another way to look at this is to consider it this way: 2 x (–5) could be the equivalent of saying “take away 5 groups of 2.” Again, this gives us (–10).

Putting all this together, we can hopefully now see why multiplying two negative numbers gives a positive number. (Keep in mind that subtracting a negative number is the same as adding a positive number.) Take the example of (–5) x (–6). We look at it as “take away 5 groups of (–6).” If I write this out, starting with nothing, I could show this as:

0 – (–6) – (–6) – (–6)- (–6)- (–6)

Of course, taking away a negative is like adding a positive, we can rewrite this as:

0 + 6 + 6 + 6 + 6 + 6 = 30

Try to think of multiplication just as a form of adding over and over again, and remember that taking away a negative is the same as adding a positive, and hopefully you will be able to make sense of it.

So, just what is a Real Number?  Quite simply, a Real Number is any number that you can express in a decimal form.  They are the numbers that you use everyday in your math homework, and which also get used in laboratories, offices, and industries.  Like I said, you don’t even realize that you’ve been using Real Numbers all along.

The set of Real Numbers is often designated with a rather fancy capital R letter, like this: 

If we think of our set of Real Numbers as being numbers that can be expressed as a decimal, we can similarly think of them as being represented on a number line.  As such, it is easy to see that Real Numbers include all the numbers on the number line, whether they are positive, negative, or zero.

Now, if we continue to think of a number line, we can further break down our set of Real Numbers into further categories.

One category is the Rational numbers, which are any numbers that can be expressed as the ratio of two integers where the denominator is not zero (such as 5 (5/1), 2/3 (0.6666…), or 0.87934 which is really 87934/100000…).  The whole numbers, represented by the ticks, are the Integers.  And the Integers themselves are are composed of Natural numbers, which are your regular ‘counting numbers’ (0, 1 apple, 2 apples, 3 apples) and their negatives.

Of course, if there are Rational numbers, we must also have Irrational numbers. Irrational numbers cannot be expressed as the ratio of two integers, and so their decimal forms extend forever without repeating.  Square roots are sometimes irrational (such as the square root of 2), as well as pi and e (if you haven’t already, you will learn about these special numbers later).

So then, in a nutshell, what is a Real Number? They are a set of numbers that can be broken down into a few categories. Rational numbers are all numbers that can be represented as a simple fraction or repeating decimal.  Irrational numbers are all numbers whose decimals continue forever without repeating. Together, the Rational and Irrational numbers fill in our number line completely, and form the set of Real Numbers.

Hopefully this makes sense and answers the question that so many students arrive here looking for: “What are Real Numbers?” Also, I hope it gives you a deeper understanding of the numbers that you have always known how to work with anyways!