[00:00:17.851] So in this module we're going to take another example of application of quadratic
equation in electrical engineering.
[00:00:27] So we in this case we're going to take two resistors, R1 and R2, and connect them in parallel.
You can see that here, R1 and R2 are connected in parallel. And we can replace this by its equivalent
resistance, R equivalent, some people also call it total resistance, but R equivalent is equal to R1 time R2
or R1 plus R2. And you will study more of that in electrical circuit course. Now, if you are given that R1 is
dependent on R2 and its value R1 then is 4 R2 plus 100 ohms, and the total value are the equivalent
value of the resistor total value going to be 100 ohms, what should be the value for R1 and R2, that is
the problem.
[00:01:22] Now, let's talk about the, before I really solve this problem, I do want to quickly talk about,
so review your what you did in physics. So the two resistors are connected in parallel. If the voltage
across them is same.
[00:01:44] So in this case, if I have this R1 and R2 and we are saying they are connected in parallel, the
voltage across that is same. So this will be, if this is a VS, this will have a VS and this will have a VS. And
the current is going to be different. So if there is a current I1 and I2 flowing through that it would be
different current.
[00:02:07] On the other hand, you will also, you may have also done that in physics, and you will do
that in circuit course if you have two resistance in series, so let me quickly draw that here,
[00:02:22] the two resistors are connected in series if you have like this. In another word, this is R1 and
R2, now, in this case, the voltage is going to be different. This is the total voltage, the voltage going to
be different, but the current is same. Ok, so here is the bottom line.
[00:02:47] If the current flowing through the two resistors, and it could be more than, more than two,
the current flowing is same through the resistors and they are connected in series. If the voltage is
same, current is different, then they are connected in parallel. Anyway, that is a quick introduction to
series and parallel combination of resistors. Ok, so let's go ahead now, solve this problem on hand
[00:03:14.769] that we have, R equivalent to be equal to R1 R2 divided by R1 plus R2.Now, we are given
the equivalent resistors will be equal to 100, right? And we are also given R1 in term of R2. So let's go
ahead and put R1 equal to 4 R2 plus 100. Ok, so that's the value of R1 multiplied by R2 and we have
denominator for R1, which is 4 R2 plus 100 plus R2. So this is what we have, so now we're left with only
R2. But let's simplify before we do anything. So let's look at the numerator. And that’s 4 R2 times R2,
which is coming out to be 4 R22 R2 times R2 is R22. Plus 100 times R2, which is 100 R2. And in the
denominator, we have 4 R2 plus R2, that gives you 5 R2 plus 100. And that is equal to 100.
[00:04:36] Now, what we want, we want equations. So we want to get it off this denominator, 5 R2 plus
100. So what do we do? We go ahead and multiply both side by this expression,
[00:04:49] 5 R2 plus 100, so let's do that, so we multiply left-hand side by this quantity and we do the
same thing to the right-hand side. We cannot just multiply on one side. Whatever we do on the lefthand side, we have to do the same thing on the right-hand side. So what do we have here, 5, this is
what we had and we multiplied by same thing, 5 R2 plus 100. And if we do that, this cancel with that,
and we left with this quantity. So what do we have here, we left with 100 times 5 R2 plus 100 equal to 4
R22 plus 100 R2. So let's simplify this, so let's multiply this thing by 100 so what we have, 500 R2 plus
10,000. Now, I could have written that in the scientific notation, 10 raised to the power of 4, but let's
leave that the way it is, so this is 4 R22 plus 100 R2.
[00:06:13] So now, what we need to do is simplify this. Let's take the whole thing, subtract it from both
side 500 R2 plus 10,000. If we subtract from this side, it becomes 0, and we have subtract the same
thing from the right-hand side, so what we have is this. Minus 500 R2, so we are subtracting this from
both side, right, so on the left-hand side became 0 and we subtracted that. So let's now simplify and
switch this around and write this on the left-hand side, 4 R22 and we have 100 R2, now when we
distribute this negative sign to both side now and that's a common mistake, I've seen this many times,
students multiplying this only with a negative sign and this leave alone, plus. You have to do both
terms. So this will be minus 500R2 minus 10,000. And that will be equal to 0. Now, simplify this
quantity, what do we have here, we have 4 R22 minus 400 R2 minus 10,000. And we have a quadratic
equation
[00:07:49]in R2 and we can solve using three method I described in example 1. But before we do that, I
want to go ahead and make the leading term to be equal to 1 also, to simplify things.
[00:08:05.515] So let's divide everything by 4, every term by 4, so if we do that, you get R22, divide this
by 4, you get 2 R22, divide 400 by 4, you get 100 R2 and divide this by 4, that will be minus 2,500 and we
have, we need to solve this quadratic equation, ok? So, let's go do that. Let's use the first method,
which is, we call factoring, we did that in the first example, and I explained this in the first example,
[00:08:49] to factor that, the product of the two factor
[00:08:54] should be 2,500. And the sum should be equal to 100 now. In this case, we had to be careful
here, this is also negative, so the product have to be minus 2,500, right? So how do we do that, how do
we get 2,500?
[00:09:19.895] When you have something like this, it become difficult to solve, also I said in the first
example, factoring becomes a problem when you do not have a whole number and so in this case, it's
going to be difficult to solve this problem using the factoring, so we will skip that and go to
[00:09:45.336] the quadratic formula. Ok, so, now what we talked about in the first example, we have
ax2 plus bx plus c, so x in this case is R2, the a term is 1, b is equal to minus 100, and C is equal to minus
2,500. Ok, ax2 plus bx, x is R2, so b will be minus 100, plus c, and c is 2,500. So let's go ahead and solve
that. So my x is equal to R2, again, minus b minus minus 100. B is minus 100, so minus b plus and minus
b2, so it's minus 100 squared minus 4 a c so it's 4 times a is equal to 1, c is equal to -2,500, square root
of this, divided by 2 a, 2 time a is simply equal to 2, alright, so let's see if we can solve that. Minus
minus is plus, so this will become 100 here, plus and minus, so 100 squared is going to be 10,000. 10 to
power of 4, ok? So you can write 10 to power of 4 or simply write 10,000. Minus minus plus again, plus
and there is 2,500 times 4, is also going to be 10,000. Square root of that, divided by 2, so what do we
have here, we have 100 plus and minus, what I'm going to do write this in the scientific notation
becomes easier to solve and 10,000 is 10 to power of 4, that's 104 , so we have 2 times 10 raised to the
power of 4, square root divided by 2, now, as you know, the 10 to power of 4, square root when you
take square root, that exponent is going to be divided by 2, so that's going to come out to be 10 raised
to the power of 2, so we have 100 plus and minus 10 raised to the power of 2 square root of 2. So we
take this, this out and what is left is square root of 2 divided by 2. Now, square root of 2, couple of
things I think
[00:12:46] you should permanently put that in your mind, is
[[00:12:52] square root of 2 is 1.414. I think remember that because you're going to use this again and
again. Similarly, square root of 3 is 1.732, ok? So always remember that will become easier to do the
math later on. Ok, so this is square root of 2 is 1.414 and if I multiply it by 100 that becomes 141.4, so
this will be hundred plus 141.4 divided by 2. This is plus and minus, ok. So now knowing that now, let's
go ahead and solve for the value of R2. If you take the plus value that will be 100 plus 141.4 divided by
2 and if you take the negative value, that will come out to be equal to 100 minus 141.4 divided by 2, so
we have two values there. So what comes out to be here that's 241., 241.4 divided by 2 will come out
to be 120.7. If you take 241.4 divided 2, it's going to come out to be 120.7. Now 100 minus 141.4 is
minus 41.4 divided by 2 is going to come out to be a negative number and that negative number will be
20.7. And these values are the ohm. Now you see why it was difficult to factor in there, first of all, the
answers, one is positive, one is negative and secondly, they are not whole numbers. So when you have
something like that, it's difficult to use the method called factoring so that's why we didn't use it. Now,
we have another problem over here. Now one answer is positive, the other answer is negative.
[00:15:14] Now you need to think about, is this possible to have a negative value for, of the resistance?
The answer is no.
[00:15:23] So really, this is not the answer, so really the value of R2 is equal to 120.7 ohm. Ok, naturally
this answer is not practical answer, only one answer, which is 121, 120.7. So, knowing R2 and we also
know, you are given R1, and that's why what we used right in the beginning, R1 is 4 R2 plus 100, so using
that value, so we can substitute this 4 times 120.7 plus 100 and we can find the value of R1, so if we
multiply that to 120 times 4, is 480 and that is going to be 2.8, so it's going to be 482.8. So 480 and the
7 times 4 is 28, so decimal points, 2.8, that comes out to be 482.8 plus 100. So the answer for R1 is
equal to 582.8 ohms. So R1 is equal to that. So we have solved the value of R1 and R2 using the
quadratic formula.
[00:17:08] We going to solve this quadratic equation now using the third method, which is completing
the square.
[00:17:16] So let's start with the quadratic equation again, R22 minus 100 R2 minus 2,500 equal to 0.
Alright, as we talked about in the previous example, let's go ahead and take the 2,500 on the other side
and leave the equation with R2 on the left-hand side. So we have R22 minus 100 R2 equal to 2,500.
[00:17:57] Alright, so, let's take the left-hand side to complete this, as a square,
[00:18:05] what we're going to take the coefficients of R2, which is 100, divided by 2, so it's 50, and
square that. So plus 50 squared and we going to do that both sides. So what do we have here? We
have R22 minus 100 R2 plus 2,500 equal to 2,500 plus 2,500 is 5,00. So if you look in the left-hand side, it
is a complete square and if you look 50 over here, square that, so this is R22 minus 2 times 50 times 1,
which gives me 100 R2, 50 squared is 2,500 and that is equal to 500, 5000. Now, let's make the righthand side also the complete square, so the 5000 if I squared that, plus gives you, plus square root of
5000, square is 5000 and negative also gives you 5000, so now let's take the square root of both sides,
so we have R2 minus 50, equal to plus and minus square root of 5000. Ok, and square root of 5000 is
equal to 70.7. So plus and minus 70.7, so R2 minus 50 is equal to plus and minus, so if I take 50 on the
other side, now R2 is equal to 50 plus and minus 70.7, alright. So let's take the plus first, so it will be 50
plus 70.7 and the other one is going to be 50 minus, let's take a negative sign, 50 minus 70.7. So
basically that gives me R2 as 120.7 and the second answer is going to be minus 20.7 ohms. Now,
[00:20:36] physically, this answer is not positive.
[00:20:40] The resistance is always positive, so the answer is simply is going to be 127, 120.7 ohms. So
the answer for this problem, by using the completing the square method, is 120.7. Same answer we
got by quadratic formula and so we have the same answer. Now, let's find out what R1 is. As you know,
R1 was given to be 4 R2 plus 100. So R1 then is 4R2 plus 100. And if I use R2 as 120.7 and add 100 to that
so this is equal to 482.7. 127, 120.7 times 4 is equal to 482.7 plus 100. So the answer for R1 should
come out to be 582.7 ohms. And so we have solved for R1 and R2 and R1 and R2 exactly comes out to
be, that should not be 7 that should be 8, point 7 time 4 should be giving you 2.8 and that should be
2.8, that should be 8 over here. So the answer for R1 is 482.8 plus 100, which is 582.8 and it got same
answer over here and R2 was 120.7,
[00:22:38.9] so we got the same answer using completing the square and quadratic formula.