Understanding Quadratic Equations Methods (vid20)

Okay, so last time we started quadratic equations We started by doing equations of this form ax squared plus C is equal to zero so there we noted that essentially this boils down to something like x squared equaling a constant and We saw that that would lead you to x equaling plus or minus the square root of k, assuming k was positive. So we did that last time. Now we're going to look into the more general case, ax squared plus bx plus c. And there are essentially two main methods to do this. One is by the method of factoring. And the other is the quadratic formula. And we're also going to provide a proof of the quadratic formula which uses the method of completing the square. So we're not going to talk in detail about completing the square, we're just going to kind of use it to prove the quadratic formula later on. So the method of factoring of course only works if you're polynomial factors, if not then you pretty much got a resort to the quadratic formula which essentially is derived from completing the square. So how do we fail? Let's try this one, let's factor it. So we have two numbers that have to multiply to negative 60, 30 times negative 2, and that have to add up to 7. and we see that 12 and minus 5 are numbers that multiply to negative 60 and add up to 7. So we rewrite 30x squared plus 12x minus 5x minus 2. is equal to zero and we factor out a 6x here and we get 5x plus 2. Factor out a minus 1 in the second set of two we get 5x plus 2. And so we have 5x plus 2 times 6x minus 1 equaling 0. So here we note that there is a general rule of thumb for real numbers that if you have a product of a times b being 0, then either a is 0 or b is 0, or sometimes both. So we're going to apply this idea and we say okay well if we have a product of two numbers at 0 that means 5x plus 2 is 0 or 6x minus 1 is 0. So from the first one we get 5x is equal to minus 2 dividing by 5 on both sides you get x equals negative 2 fifths or from this one we get 6x equals 1 so dividing by 6 you get x equals 1 over 6 so these are your two possible these are your two solutions to this equation okay now we're going to go to the quadratic formula which we're just going to state first without without proving it we'll prove it at the end So if you're given ax squared plus bx plus c, and you ask what values of x solve this equation, Well, it would be x equals negative b plus or minus the square root of b squared minus 4ac over 2a. So these are actually two solutions. Two solutions unless b... squared minus 4ac is 0. There's a couple of situations where we don't get solutions or sometimes we only get one solution. But in general, you have two solutions. So it's negative b plus the square root of b squared minus 4ac. over 2a, this is one solution, as well as the same thing except a negative sign in between the negative b and the square root of b squared minus 4ac. So these are the two solutions. here note we're gonna we can check so if we had to factor well we'd have to find two numbers that multiply to negative 7 and then add up to 4 well the only numbers that multiply to 7 or 7 and 1 so it would be either 7 and negative 1 or negative 1 and 7 neither of those pairs add up to 4 so this is not factorable So we can't use the method of factoring. What we can use then is the quadratic formula. And we realize here in this case a is 1, b is 4, c is minus 7. So x then equals negative b, which is negative 4, plus or minus the square root of b squared minus 4ac all over 2a. So this is negative 4 plus or minus the square root of 16 plus 28 all over 2. So this is negative 4 plus or minus the square root of 44 over 2. This is 4 times 11. So this is negative 4 plus or minus the square root of 4. times 11 over 2. Negative 4 plus or minus 2 rad 11 over 2. And then we can cancel this is we can factor out the 2 on the top. and we get negative 2 plus rad11 divided by 2 and so we can cancel these guys and so our final answer is x equals negative 2 plus or minus square root of 11 or written separately negative 2 plus rad11 or x equals negative 2 minus rad11. but for compactness. We usually just write it with a plus or minus. And that's it. That's the method of the quadratic formula. Okay, so the question then comes, how do we know that the quadratic formula actually works? So this is... a general proof. So suppose we have ax squared plus bx plus c being zero. How do we solve this? Well what we're going to do is we're going to bring the c over a or we're going to fact well let's factor out the a first. So we're going to factor out a and we have x squared plus b over ax and we're going to leave this c on the other side. Okay. So what we're going to do is we're going to essentially complete the square on this. So what we do is we take a half of b over a and we're going to square that. So that's b squared over 4a squared. And so we have a times x squared plus b over ax plus b squared over 4a squared. Now if we added this, because we... we added this we have to subtract it because it wasn't there in the first place. The expression here has to be the same. This has to be the same as the original expression. So this is what I have. And this thing right here is the construction of completing the square. You take a half of the middle term and you square it. So the middle term here in this particular case is b over a. We took a half of that and we squared it. And that's what we add and subtract. into this particular expression. So we have a and then we have x squared plus b over ax plus b squared over 4a squared. And I'm going to close the parentheses here. This other term right here is going to be a times, it's going to become a multiplied by that particular term. So I get minus b squared over 4a because this a on which I'm multiplying cancels with one of the a's in the denominator here. And this equals minus c. So now we have a and we we can um this and the on the left hand side becomes x plus b over 2a quantity squared. So this piece here factors And I'll leave you to check that this actually is true, that this piece is identically equal to this piece. So this now, this whole thing, minus b... squared over 4a is equal to minus c. So what we're going to do now, just to clear the denominator, we're going to multiply the entire left and right hand side. hand side by 4a just to clear the denominators. So what we get is 4a squared times x plus b over 2a squared minus b squared is equal to minus 4ac. And now you can see that this negative 4ac and the b squared are coming together. So what we have is 4a squared times x plus b over 2a squared is equal to b squared minus 4ac. And now we're going to divide by 4a squared. So x plus b over 2a quantity squared is equal to b squared minus 4ac all over 4a squared. So what are we left with? Well, now... Remember if we have a perfect square here, think of this as like y squared, if we had y squared equaling b squared minus 4ac over 4a squared, then y is plus or minus the square root. of b squared minus 4ac over 4a squared. Okay, well in this case y is x plus b over 2a, right? Because if I substitute y for x plus b over 2a, I have y squared on the left and this expression on the right. So this is my solution. But I want everything in terms of x. So I go back to x and I realize this is x plus b over 2a. equaling plus or minus square root of b squared minus 4ac. And on the denominator here, the square root of 4a squared is just 2a. So what I realize is x is actually equal to, and I now bring this piece over to this side by subtracting b over 2a on both sides. I get negative b over 2a plus or minus square root of b squared minus 4ac over 2a. and now we have the same denominator. So indeed, this is just negative b plus or minus square root of b squared minus 4ac all over 2a. And that's our final quadratic formula. So essentially the quadratic formula is an output, outcome of the process of completing the square, which is pretty much where you're taking half of a middle term and squaring it and then adding and subtracting. So that's the procedure. Okay, so this is the proof of the quadratic formula. We're going to do maybe one more example of the quadratic formula in the next video, and then we'll do an application of solving quadratic equations as well. All right, that's it for now.

And there are essentially two main methods to do this. One is by the method of factoring. And the other is the quadratic formula. And we're also going to provide a proof of the quadratic formula which uses the method of completing the square.

So we're not going to talk in detail about completing the square, we're just going to kind of use it to prove the quadratic formula later on. So the method of factoring of course only works if you're polynomial factors, if not then you pretty much got a resort to the quadratic formula which essentially is derived from completing the square. So how do we fail? Let's try this one, let's factor it. So we have two numbers that have to multiply to negative 60, 30 times negative 2, and that have to add up to 7. and we see that 12 and minus 5 are numbers that multiply to negative 60 and add up to 7. So we rewrite 30x squared plus 12x minus 5x minus 2. is equal to zero and we factor out a 6x here and we get 5x plus 2. Factor out a minus 1 in the second set of two we get 5x plus 2. And so we have 5x plus 2 times 6x minus 1 equaling 0. So here we note that there is a general rule of thumb for real numbers that if you have a product of a times b being 0, then either a is 0 or b is 0, or sometimes both.

So we're going to apply this idea and we say okay well if we have a product of two numbers at 0 that means 5x plus 2 is 0 or 6x minus 1 is 0. So from the first one we get 5x is equal to minus 2 dividing by 5 on both sides you get x equals negative 2 fifths or from this one we get 6x equals 1 so dividing by 6 you get x equals 1 over 6 so these are your two possible these are your two solutions to this equation okay now we're going to go to the quadratic formula which we're just going to state first without without proving it we'll prove it at the end So if you're given ax squared plus bx plus c, and you ask what values of x solve this equation, Well, it would be x equals negative b plus or minus the square root of b squared minus 4ac over 2a. So these are actually two solutions. Two solutions unless b... squared minus 4ac is 0. There's a couple of situations where we don't get solutions or sometimes we only get one solution. But in general, you have two solutions.

So it's negative b plus the square root of b squared minus 4ac. over 2a, this is one solution, as well as the same thing except a negative sign in between the negative b and the square root of b squared minus 4ac. So these are the two solutions.

here note we're gonna we can check so if we had to factor well we'd have to find two numbers that multiply to negative 7 and then add up to 4 well the only numbers that multiply to 7 or 7 and 1 so it would be either 7 and negative 1 or negative 1 and 7 neither of those pairs add up to 4 so this is not factorable So we can't use the method of factoring. What we can use then is the quadratic formula. And we realize here in this case a is 1, b is 4, c is minus 7. So x then equals negative b, which is negative 4, plus or minus the square root of b squared minus 4ac all over 2a. So this is negative 4 plus or minus the square root of 16 plus 28 all over 2. So this is negative 4 plus or minus the square root of 44 over 2. This is 4 times 11. So this is negative 4 plus or minus the square root of 4. times 11 over 2. Negative 4 plus or minus 2 rad 11 over 2. And then we can cancel this is we can factor out the 2 on the top.

and we get negative 2 plus rad11 divided by 2 and so we can cancel these guys and so our final answer is x equals negative 2 plus or minus square root of 11 or written separately negative 2 plus rad11 or x equals negative 2 minus rad11. but for compactness. We usually just write it with a plus or minus.

And that's it. That's the method of the quadratic formula. Okay, so the question then comes, how do we know that the quadratic formula actually works?

So this is... a general proof. So suppose we have ax squared plus bx plus c being zero. How do we solve this? Well what we're going to do is we're going to bring the c over a or we're going to fact well let's factor out the a first.

So we're going to factor out a and we have x squared plus b over ax and we're going to leave this c on the other side. Okay. So what we're going to do is we're going to essentially complete the square on this.

So what we do is we take a half of b over a and we're going to square that. So that's b squared over 4a squared. And so we have a times x squared plus b over ax plus b squared over 4a squared. Now if we added this, because we... we added this we have to subtract it because it wasn't there in the first place.

The expression here has to be the same. This has to be the same as the original expression. So this is what I have.

And this thing right here is the construction of completing the square. You take a half of the middle term and you square it. So the middle term here in this particular case is b over a.

We took a half of that and we squared it. And that's what we add and subtract. into this particular expression.

So we have a and then we have x squared plus b over ax plus b squared over 4a squared. And I'm going to close the parentheses here. This other term right here is going to be a times, it's going to become a multiplied by that particular term. So I get minus b squared over 4a because this a on which I'm multiplying cancels with one of the a's in the denominator here. And this equals minus c.

So now we have a and we we can um this and the on the left hand side becomes x plus b over 2a quantity squared. So this piece here factors And I'll leave you to check that this actually is true, that this piece is identically equal to this piece. So this now, this whole thing, minus b... squared over 4a is equal to minus c.

So what we're going to do now, just to clear the denominator, we're going to multiply the entire left and right hand side. hand side by 4a just to clear the denominators. So what we get is 4a squared times x plus b over 2a squared minus b squared is equal to minus 4ac.

And now you can see that this negative 4ac and the b squared are coming together. So what we have is 4a squared times x plus b over 2a squared is equal to b squared minus 4ac. And now we're going to divide by 4a squared.

So x plus b over 2a quantity squared is equal to b squared minus 4ac all over 4a squared. So what are we left with? Well, now...

Remember if we have a perfect square here, think of this as like y squared, if we had y squared equaling b squared minus 4ac over 4a squared, then y is plus or minus the square root. of b squared minus 4ac over 4a squared. Okay, well in this case y is x plus b over 2a, right? Because if I substitute y for x plus b over 2a, I have y squared on the left and this expression on the right.

So this is my solution. But I want everything in terms of x. So I go back to x and I realize this is x plus b over 2a.

equaling plus or minus square root of b squared minus 4ac. And on the denominator here, the square root of 4a squared is just 2a. So what I realize is x is actually equal to, and I now bring this piece over to this side by subtracting b over 2a on both sides.

I get negative b over 2a plus or minus square root of b squared minus 4ac over 2a. and now we have the same denominator. So indeed, this is just negative b plus or minus square root of b squared minus 4ac all over 2a.

And that's our final quadratic formula. So essentially the quadratic formula is an output, outcome of the process of completing the square, which is pretty much where you're taking half of a middle term and squaring it and then adding and subtracting. So that's the procedure. Okay, so this is the proof of the quadratic formula. We're going to do maybe one more example of the quadratic formula in the next video, and then we'll do an application of solving quadratic equations as well.

All right, that's it for now.

Transcript for:Understanding Quadratic Equations Methods (vid20)

Transcript for:
Understanding Quadratic Equations Methods (vid20)