Solve by completing the square: Integer solutions | Algebra I | Khan Academy

So we're given this equation here. What I want you to do is pause this video and see if you can solve it. What x values satisfy the equation?

All right, now let's work through this together. One technique could be just let's just try to complete the square here on the left-hand side. So to do that, let me write it this way: x squared minus 8x and then I have, all right, the plus 1 out here is equal to 85.

Now, if I want to complete the square, I just have to think, what can I add to both sides of this equation that could make this part of the left-hand expression a perfect square? Well, if I look at this negative 8 coefficient on the first degree term, I could say, okay, let me take half of negative 8. That would be negative 4, and then negative 4 squared is going to be positive 16.

So I'm going to add a positive 16 on the left-hand side, and if I want, I could then subtract a 16 from the left-hand side or I could add a 16 on the right-hand side. Notice I've just done the same thing to both sides of this equation. Why was that useful? Well, now what I've just put in parentheses is a perfect square. This is the same thing as x minus 4 squared. It was by design; we looked at that negative 8, half of that is negative 4. You square it, you get 16.

And you can verify x minus 4 times x minus 4 is indeed equal to this. And then we have plus 1 is going to be equal to what's 85 plus 16? That is 101. Now we want to get rid of this one on the left-hand side, and the easiest way we can do that is subtract 1 from both sides. That way we'll just isolate that x minus 4 squared, and we are left with x minus 4 squared is going to be equal to 100.

Now, if something squared is equal to 100, that means that the something is equal to the positive or the negative square root of 100. Or that, that something x minus 4 is equal to positive or negative, positive or negative 10. All I did is took the plus or minus square root of 100, and this makes sense. If I took positive 10 squared, I'll get 100. If I take negative 10 squared, I get 100.

So x minus 4 could be either one of those. And now I just add 4 to both sides of this equation. What do I get? I get x is equal to 4 plus or minus 10. Or another way of thinking about it, I could write it as x is equal to 4 plus 10, which is 14, and then 4 minus 10 is equal to negative 6.

So these are two ways to solve it, but there's other ways to solve this equation. We could, right from the get-go, try to subtract 85 from both sides. Some people feel more comfortable solving quadratics if they have the quadratic expression be equal to zero. If you did that, you would get x squared minus 8x minus 84 is equal to 0.

All I did is I subtracted 85 from both sides of this equation to get this right over here. Now this one we can approach in two different ways. We can complete the square again, or we could just try to factor. If we complete the square, we're going to see something very similar to this. Actually, let me just do that really fast. If I look at this part right over there, I could say x squared minus 8x, and then once again half of negative eight is negative four; that squared is plus sixteen, and then I'd have minus eighty-four.

Let me do that in that blue color so we can keep track: minus 84. If I added 16 on the left-hand side, I could either add that to the right-hand side so both sides have 16 added to them, or if I want to maintain the equality, I could just subtract 16 from the left-hand side. So I've added 16, subtracted 16; I haven't changed the left-hand side's value, and then that would be equal to 0.

This part right over here, this is x minus 4 squared. This part right over here is minus 100 is equal to zero, and then you add 100 to both sides of this and you get exactly this step right over here.

Now, another way that we could have approached it without completing the square, we could have said x squared minus 8x minus 84 is equal to zero and think about what two numbers, if I multiply them, I get negative 84. So they'd have to have different signs since when I take their product I get a negative number, and when I add them together I get negative 8.

And there we could just look at the factorization of negative 84. Of 84 generally, it could be 2 times... let's just think about 84. 84 could be 2 times 42, and obviously, one of them would have to be negative; one of them would have to be positive in order to get to negative 84. But the difference between these two numbers, if one was positive and one is negative, is a lot more than eight, so that doesn't work.

So let's try... let's see, I'll do a few in my head: three times 28, but still, that difference is way more than eight. Four times... four times, let's see, four times 21. Now that difference between 4 and 21 is still larger than 8. Let's see, 5 doesn't go into it; 6 times 14. That's interesting; now, okay, so let's think about this.

So 6 times 14 is equal to 84. One of them has to be negative, and since our—when we take the sum of the two numbers—we get a negative number, that means the larger one is negative. So let's see, 6 times negative 14 is negative 84; 6 plus negative 14 is indeed equal to negative eight. So we can factor this as x plus 6 times x minus 14 is equal to zero.

And so the product of two things is equal to zero; that means what? If either of them is equal to zero, that would make the entire expression equal to zero. So we could say x plus 6 is equal to zero, or x minus 14 is equal to zero. Subtract 6 from both sides here; we get x is equal to negative 6, or add 14 to both sides here, or x is equal to 14. Exactly what we got up here.