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Completing the Square

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Hi. My name is Anthony Varela. And today, we're going to talk about completing the square. So we're going to look at a quadratic expression and focus on the x-term coefficient and the constant term.

Through algebraic manipulation, we'll develop a perfect square trinomial. So we'll talk about that in a minute. And this is all going to involve what we call completing the square to rewrite and solve a quadratic equation.

So before we get into the nuts and bolts, I'd like to go through a quick example of solving a quadratic equation in a very specific form. So we have x plus 4. So it's a binomial. And we're squaring that quantity. And let's say that equals 2. How would we solve for x?

Well, just using our inverse operations, we could take the square root of both sides of this equation. So that undoes the quantity squared on this side. I always remember to use plus or minus in equations. So we have plus or minus the square root of 2. And to isolate x, we'll just subtract a 4 from both sides.

So x is negative 4 plus or minus the square root of 2. So this would actually be two x-values. One is negative 4 plus square root of 2. And the other is negative 4 minus the square root of 2.

Well, that was a pretty easy equation to solve for. So wouldn't it be nice if all quadratic equations could be written as a binomial squared? Well, unfortunately, they don't all come that way. But using completing the square, we might be able to write it as such.

So to develop this process for completing the square, I'd like to relate this back to FOIL. So if we have a general binomial squared-- so x plus a quantity squared-- and we were to FOIL this out-- so multiplying the first, outside, inside, last terms in x plus a times x plus a-- we would get x squared plus 2ax plus a squared.

Now, pay particular attention to the coefficient in front of x and our constant term. We have 2a and a squared. So notice that if we take half of the x-term coefficient and square it, we will get the constant term.

So let's look at a more specific example, x plus 3, quantity squared. If we were to FOIL this out, we would have x squared plus 6x plus 9. And notice if we take 6 and cut it in half, we get 3. And if we square 3, we get 9.

So here we have then our binomial squared-- so x plus a, quantity squared. This equals x squared plus 2ax plus a squared. And this is actually our perfect square trinomial I mentioned earlier-- one, two, three terms. So it's a trinomial. And we can rewrite it as a binomial squared. So that's what we'd like to do when we complete the square.

So completing the square follows a very specific process. So we'll go through that in the following example. So if we have y equals 2x squared plus x minus 2. And we'd like to solve this quadratic equation.

We recall that solving a quadratic equation means finding its roots, its zeros, its values that make y equal 0. So I'm just going to put in 0. We'd like to solve for x when 2x squared plus x minus 2 equals 0.

So the first step to completing the square is that we want to move the constant term. And so I mean move that to the other side of the equation. So we have a minus 2. So I'm going to add 2 to both sides of the equation.

So what we have is 2 equals 2x squared plus x. So we have no constant term. Everything on this side of the equation has a factor of x.

The next step is to divide the entire equation by the x squared coefficient. So in this case, it is 2. So we're going to divide this by 2, divide this by 2, divide this by 2. So what do we get? Well, we get 1 equals x squared plus 1/2 x.

So our next step then is we're going to take that x-term coefficient. So in this case, it's 1/2. We are going to first divide it by 2. So 1/2 divided by 2 is 1/4. And then we'll square it. So cut in half and then square it.

So now what do we do then with this 1/4 squared? Well, we're going to add it to both sides of the equation. So I've added 1/4 squared here and 1/4 squared here.

So let's just clean up our equation so far. We have x squared plus 1/2 x plus 1/16. That would be 1/4 times 1/4. And when we add 1/16 and 1 on this side of the equation, we get 17 over 16.

Well, now, oddly enough, we have a perfect square trinomial. This actually follows this form right here. So we can rewrite it as x plus a quantity squared. We just need to identify what a is. Well, it's going to be this 1/4. So I can write this then as x plus 1/4, quantity squared. And that equals 17/16.

So oddly, enough we started with this equation, 0 equals 2x squared plus x minus 2. And we've rewritten it as 17/16 equals x plus 1/4, quantity squared by following this completing the square process.

So now we have something that looks like our example from the beginning of this video. And we can solve for x by taking the square root of both sides of my equation. So that eliminates the quantity squared here. I have plus or minus the square root of 17 over 16.

I can subtract a 1/4 from both sides of the equation. So although this looks like a mess in our calculator, if we type in negative 1/4 plus the square root of 17 over 16 and then negative 1/4 minus the square root of 17 over 16, we'll get two x-values. One will be negative 1.28 when we round. And the other will be 0.78 when we round. So those are our two solutions then to the quadratic equation.

So let's review completing the square. Well, we used a binomial squared expanded-- so this is a perfect square trinomial-- to think of this relationship between the x-term coefficient and the constant term. Completing the square is the process of converting a quadratic equation in the form ax squared plus bx plus c into an expression involving a perfect square trinomial.

And here are the steps for completing the square. You can follow them in order. And you should have no trouble solving quadratic equations using this completing the square method.

So thanks for watching this tutorial and completing the square-- hope to see you next time.