The Numerist -

Time 2020-12-31 12:03:47

Web Name: The Numerist -

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So, now that you know the answer to ‘What is the Quadratic Formula,’ next I will show you examples of using it. Refer back to my last post to familiarize yourself with what the quadratic formula looks like. I’ve also explained there the nature of the roots of a quadratic equation. If you haven’t read it, I recommend taking a look as it might help you to visualize and to find the solution to a quadratic equation easier.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 = 0Comparing 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 = 4xIt’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 (specifically, Grouping in my Methods of Factoring post), 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 = 4x2x3 + 3x2 4x = 0x(2x2 + 3x 4) = 0Looks 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.Following my posts on How to Solve Quadratic Equations (here and here), you will soon find that not all quadratic equations can be solved by quadratic factoring, and you will come to rely on The Quadratic Formula to help you. As a quick refresher, a quadratic equation is one which takes the form of ax2 + bx + c = 0, as long as the “a” term is not zero. In other words, a quadratic equation is one in which there is an x2. (The “b” or “c” term can be zero.) I have already described the process you should follow if your question can be factored down, and you can express it as a product of two smaller expressions. Then, you can solve for two roots by letting each of the small expressions equal zero. I highly recommend reading my previous post if you need to go over this quadratic factoring technique.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….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 = 0First, 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!(x + 2)(x + 5) = 0This 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 = 0And 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 = 04 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. 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 0See? That doesn’t sound so bad.

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