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Completing the Square

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Today we're going to talk about completing the square. Completing the square is one method that you can use to solve a quadratic equation. So we'll go over the idea behind the method of completing the square, and then we'll do some examples.

So let's start by looking at how we would solve an equation that's in this form, for example the equation x plus 3 squared equals 25. I'm going to solve this by using inverse operations to isolate the x variable.

And the first inverse operation that I would use would be one to cancel my 2 exponent. And that's going to be the square root operation. Now x plus 3 squared equals 25 means that x plus 3 could be either 5 or negative 5, because both 5 and negative 5 squared equal 25. So when I take the square root of both sides, here I know that those two operations will cancel, so I'll be left with x plus 3. And I know that would be equal to either a positive or negative 5.

So to further solve this, I'm going to write this as two equations that we can solve to get our two different answers. So x plus 3 equals 5, and x plus 3 equals negative 5. For my first equation, to get x by itself, I'm going to subtract 3 from both sides. So this will cancel, and I'll be left with x is equal to 5 minus 3 or 2. So x equals 2 is my first answer.

And then for the second equation, I'll also subtract 3 on both sides to isolate x. And now I'm left with x is equal to negative 5 minus 3, or negative 8.

So we've seen how to solve a quadratic equation, such as x plus 3 squared equals 25, or any equation in the form x plus a squared equals y, where a and y are real numbers. But not all quadratic equations come in this form. So we use a process called completing the square to rewrite a quadratic equation, any quadratic equation in this form, so that it can be solved using the process that we did for this last example.

So before we do examples using completing the square, let's look again at our relationship between the factored form of a quadratic expression and the expanded form. So x plus a squared, in factored form, can also be written as x plus a times x plus a, which is also in factored form. And we can write this in expanded form using FOIL.

So this would give me x squared plus a times x plus another a times x plus a times a, or a squared. And I can combine these two terms to give me x squared plus 2ax plus a squared. So I can see that this in expanded form is the same as this in factored form, which means that I can take any quadratic expression written in this form and write it in this form. So let's look at the relationship between these two terms in our expanded form.

If I looked at the coefficient of my x term, which is 2a, and I divide by 2, that would just give me a. And if I were to square that to give me a squared, I'd see that's the same as my constant term. So let's see what that would look like, using an example with numbers.

Let's say, I have x plus 3 squared. I can write this in expanded form, by writing it first as x plus 3 times x plus 3. And then using FOIL to multiply, that will give me x squared plus 3x plus another 3x plus 9. Combining my two middle terms, that gives me x squared plus 6x plus 9. So again, looking at my relationship between these two terms, if I take my coefficient for my x term 6 and I divide that by 2, that will give me three. If I then take 3 and square it, that will give me 9, which is to say as my constant term.

So now let's do some examples using completing the square. So I have the quadratic equation y is equal to 2x squared plus 8x plus 7. I want to solve this using the completing the square method, which is just another method of solving a quadratic equation in addition to the factoring method or using the quadratic formula.

So I know that to solve a quadratic equation, I'm finding the value or values of x, when y is equal to 0. So I'm going to rewrite this equation as 0 is equal to 2x squared plus 8x plus 7. To use the completing the square method, I need to start by canceling out my constant term on this side of the equation. I'm going to do that by subtracting 7 on both sides, which will give me negative 7 is equal to 2x squared plus 8x.

The second step in using the completing the square method is to make sure that the coefficient in front of your x squared term is 1. Since it's not, I need to make this to be a 1 or cancel it out by dividing by 2. And since I do that here, I need to divide each term by 2 in my equation, to keep it balanced.

Dividing by 2 here will cancel out, and I'm left with x squared. This will give me 4x. And on this side, I'll have negative 3.5. So now, I want to be able to write this expression on this side of my equation as a perfect square trinomial, which means I want to have a constant that is equal to the square of 1/2 of my x term coefficient.

So I'm going to take 1/2 of 4 or 4 divided by 2, which will give me 2. And then I'm going to square that. 2 squared will give me 4.

So I want to be able to have a constant term here of positive 4. But by writing or adding a 4 on this side of the equation, I need to keep the equation balanced, by adding 4 on the other side. So now my equation becomes 0.5 is equal to x squared plus 4x plus 4.

And now I can rewrite this in factored form as a perfect square. So I have 0.5 is equal to x plus 2 times x plus 2, because this will multiply to be x squared plus 4x plus 4, which is what I had here. I can write this as 0.5 is equal to x plus 2 squared. And now I have an equation that I can solve, using inverse operations to isolate x.

So I'll start by canceling out my 2 exponent, by taking the square root of both sides. Two of these will cancel, and I'm left with x plus 2. And that's going to be approximately equal to positive and negative 0.71.

So now I have two equations that I can solve. Positive 0.71 is approximately equal to x plus 2. And negative 0.71 is approximately equal to x plus 2. Isolating x on both sides, in this equation, I find that x is going to be approximately equal to negative 1.29, for my first solution. And subtracting 2 on both sides here, I find that x is approximately equal to negative 2.71, for my second solution.

So let's go over our key points from today. Completing the square is a method of solving a quadratic equation, in addition to factoring in the quadratic formula. The process of completing the square involves rewriting a quadratic equation as a perfect square trinomial so that it can be written in factored form as y equals x plus a squared, where a and y are real numbers.

So I hope that these key points and examples helped you understand a little bit more about completing the square. Keep using your notes, and keep on practicing. And soon you'll be a pro. Thanks for watching.