The Quadratic Formula - Video Tutorials & Practice Problems

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The quadratic formula can be used to solve for the variable x when given an algebraic equation in the form of:ax^{2} + bx - c.

The Quadratic Formula

The quadratic formula is most commonly used for questions dealing with chemical equilibrium where you have to use an ICE Chart.

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Quadratic Formula

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So here we are dealing with the quadratic formula. We are going to say the quadratic formula is used for algebraic equations that x to the 2 plusbxplusc. Okay? So here, a and b are just numerical values. Okay. So they're actual numbers. They are saying here oh and same thing with c as well. X's are just missing variables. We're going to see the quadratic formula, which is shown here, it's x equals negative b plus or minus square root of b squared minus 4ac over 2a. Here it says the quadratic formula is most commonly used for questions dealing with chemical equilibrium. When we get to the sections dealing with chemical equilibrium, we're gonna use things that are called ICE charts. And these ICE charts help me to find equilibrium concentrations. So equilibrium amounts or concentrations, that's when we typically use these. Now you may use the quadratic formula before then within your lab. So it's still important to remember. So we're gonna say although the presence of the positive negative sign gives 2 possible values for x, only one of will be significant in used as the answer. So again, because there's a plus or minus here, that means we have a possibility of 2 answers for our x. When we do different types of questions it usually means that we have only one possibility when it comes to our final answer. Now knowing this, let's see if we can solve this question here. It says, using the quadratic formula, solve for x when given the following algebraic expression. So let me take myself out of the image guys because we need room to do this. So we're gonna say here we have 4x equals negative 2.13 times 10 to the negative 4 plus 1 point 7 5 times 10 to the negative 5 over x. Again, we want our expression to look something like this. And we want at least to have an x that is squared to be able to do the quadratic formula. Here we don't have that. And we have this number over x. So what we're going to do here is we're going to multiply both sides by x. And when we do that we are going to get 4x squared equals negative 2.13 times 10 to the negative 4x. This x will cancel out with this x, so we're gonna get 1.7 plus 1.75 times 10 to the negative 5. Alright. Now we are going to say that this x variable has the larger power so it's our lead term. That means that everything has to be moved over to its side. So we are going to add 2.13 times 10 to the negative 4x to both both sides. Then we're going to subtract 1.75x10-five from both sides. So here what we're going to get at the end for our expression is 4x squared plus 2.13 times 10 to the negative 4x minus 1.75 times 10 to the negative 5. So this represents my a, my b, and my c. So my quadratic formula is remember negative b plus or minus go up so you can see it. Plus or minus b squared minus 4ac, square root of b squared minus 4ac over 2a. Alright so here the sign of b is positive, so when I throw it in it's gonna become negative now. Plus or minus b squared, -four for a, the value in front of a is a 4, and then c, the value is this, do not forget the negative sign. Okay? So if there was a negative sign here for b, let's say that this was negative, then this would have been positive, would have been the opposite, and then we would have had to put the negative there as well. K? So don't forget the signs get introduced as well. Divided by 2 times 4. Alright. So oops. So now we're gonna say that I'm gonna take the square root of everything in here. K. So what I'm gonna do first, I'm gonna figure out what everything is in there. Now if you do it correctly in your calculator, you should get square root of 2.80 times 10 to the negative 4 divided by 8. Then, realize you are going to get 2x variables because remember it is plus or minus. So x here equals -2.13 times 10 to the negative 4 plus so when I take the square root of that number it is going to give me 0.016735 which gets divided by 8. So that'll give me 0.002065 for x. And then remember we can also have x equals negative 2.13 times 10 to the negative 4 minus, because remember it's plus or minus, divided by 8. And in this case, x equals negative.002119. So these are my two answers that x could be. Again, when we finally use this, within our lectures, we're gonna be using it for usually equilibrium questions, so chemical equilibrium questions. And only one of these answers will be the vital answer. And usually it's the positive answer is the one that we use. We discard the negative one. But we'll talk about that later on when we're dealing with chemical equilibrium, ice charts, and all that stuff. But for now just remember what the quadratic formula bx plus bx plus c to give us the variables that we need in order to solve for x. We'll get 2 answers and usually only one of them is a viable answer. The other one will be discarded. We will talk about the conditions and reasons for discarding which one later on.

Even though the quadratic formula has a +/- sign that gives two answers for the variable x, only one of them will be the correct answer. You will learn how to determine the correct answer from the two possibilities.

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