Worked example: completing the square (leading coefficient ≠ 1)

We're asked to complete the square to solve 4x squared plus 40x minus 300 is equal to 0. So let me just rewrite it. So 4x squared plus 40x minus 300 is equal to 0. So just as a first step here, I don't like having this 4 out front as a coefficient on the x squared term. I'd prefer if that was a 1. So let's just divide both sides of this equation by 4. So let's just divide everything by 4. So this divided by 4, this divided by 4, that divided by 4, and the 0 divided by 4. Just dividing both sides by 4. So this will simplify to x squared plus 10x. And I can obviously do that, because as long as whatever I do to the left hand side, I also do the right hand side, that will make the equality continue to be valid. So that's why I can do that. So 40 divided by 4 is 10x. And then 300 divided by 4 is what? That is 75. Let me verify that. 4 goes into 30 seven times. 7 times 4 is 28. You subtract, you get a remainder of 2. Bring down the 0. 4 goes into 20 five times. 5 times 4 is 20. Subtract zero. So it goes 75 times. This is minus 75 is equal to 0. And right when you look at this, just the way it's written, you might try to factor this in some way. But it's pretty clear this is not a complete square, or this is not a perfect square trinomial. Because if you look at this term right here, this 10, half of this 10 is 5. And 5 squared is not 75. So this is not a perfect square. So what we want to do is somehow turn whatever we have on the left hand side into a perfect square. And I'm going to start out by kind of getting this 75 out of the way. You'll sometimes see it where people leave the 75 on the left hand side. I'm going to put on the right hand side just so it kind of clears things up a little bit. So let's add 75 to both sides to get rid of the 75 from the left hand side of the equation. And so we get x squared plus 10x, and then negative 75 plus 75. Those guys cancel out. And I'm going to leave some space here, because we're going to add something here to complete the square that is equal to 75. So all I did is add 75 to both sides of this equation. Now, in this step, this is really the meat of completing the square. I want to add something to both sides of this equation. I can't add to only one side of the equation. So I want to add something to both sides of this equation so that this left hand side becomes a perfect square. And the way we can do that, and saw this in the last video where we constructed a perfect square trinomial, is that this last term-- or I should say, what we see on the left hand side, not the last term, this expression on the left hand side, it will be a perfect square if we have a constant term that is the square of half of the coefficient on the first degree term. So the coefficient here is 10. Half of 10 is 5. 5 squared is 25. So I'm going to add 25 to the left hand side. And of course, in order to maintain the equality, anything I do the left hand side, I also have to do to the right hand side. And now we see that this is a perfect square. We say, hey, what two numbers if I add them I get 10 and when I multiply them I get 25? Well, that's 5 and 5. So when we factor this, what we see on the left hand side simplifies to, this is x plus 5 squared. x plus 5 times x plus 5. And you can look at the videos on factoring if you find that confusing. Or you could look at the last video on constructing perfect square trinomials. I encourage you to square this and see that you get exactly this. And this will be equal to 75 plus 25, which is equal to 100. And so now we're saying that something squared is equal to 100. So really, this is something right over here-- if I say something squared is equal to 100, that means that that something is one of the square roots of 100. And we know that 100 has two square roots. It has positive 10 and it has negative 10. So we could say that x plus 5, the something that we were squaring, that must be one of the square roots of 100. So that must be equal to the plus or minus square root of 100, or plus or minus 10. Or we could separate it out. We could say that x plus 5 is equal to 10, or x plus 5 is equal to negative 10. On this side right here, I can just subtract 5 from both sides of this equation and I would get-- I'll just write it out. Subtracting 5 from both sides, I get x is equal to 5. And over here, I could subtract 5 from both sides again-- I subtracted 5 in both cases-- subtract 5 again and I can get x is equal to negative 15. So those are my two solutions that I got to solve this equation. We can verify that they actually work, and I'll do that in blue. So let's try with 5. I'll just do one of them. I'll leave the other one for you. I'll leave the other one for you to verify that it works. So 4 times x squared. So 4 times 25 plus 40 times 5 minus 300 needs to be equal to 0. 4 times 25 is 100. 40 times 5 is 200. We're going to subtract that 300. 100 plus 200 minus 300, that definitely equals 0. So x equals 5 worked. And I think you'll find that x equals negative 15 will also work when you substitute it into this right over here.