Tag Archives: methods of factoring

How to Solve Quadratic Equations – Part 2

This post continues from where my last post left off, on how to solve quadratic equations. I explained the general form that a quadratic equation will take, with the key being that there is an x2 term present. To solve them without using the quadratic formula, you need to use a bit of factoring methods to come up with the roots. In particular, one common factoring method to use is the grouping method of factoring. Then, once factored, you consider the property that says “two terms multiplied will equal zero only if one or both of those terms is 0.” This may seem like a lot of work, and may sound a bit confusing with all the steps you need to take. But I think with a bit of practice you will come to better appreciate and understand the process you need to follow to arrive at your solution. You will see that you already know the individual steps you need to solve the equation. You just need to become familiar with the order that you use these steps.

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.

How to Solve Quadratic Equations

Following up my previous post that gave you advice on how to solve equations, in this post I would like to go over some strategies on how to solve quadratic equations. Quadratic equations become very common in high school math and college math, and they require a bit more work sometimes to solve. You may already have experience using the quadratic formula, which I will explain shortly and is extraordinarily good to memorize! First though, let’s go over solving quadratic equations. To do this, you will commonly rely on factoring quadratics techniques. You can refer to my previous post on methods of factoring for some additional tips!

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.

More to come in my next post, How To Solve Quadratic Equations – Part 2….

Methods of Factoring

For this post, I would like to do one that talks about various methods of factoring which you can use to find factors. There are several different factoring techniques that you can use, depending on the situation, so I think that I would like to include a variety of methods or factoring tricks.

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).