Topic A: Lessons 6-7: Solving basic one-variable quadratic equations
We're asked to solve the equation 2x squared plus 3 is equal to 75. So in this situation, it looks like we might be able to isolate the x squared pretty simply. Because there's only one term that involves an x here. It's only this x squared term. So let's try to do that. So let me just rewrite it. We have 2x squared plus 3 is equal to 75. And we're going to try to isolate this x squared over here. And the best way to do that, or at least the first step, would be to subtract 3 from both sides of this equation. So let's subtract 3 from both sides. The left hand side, we're just left with 2x squared. That was the whole point of subtracting 3 from both sides. And on the right hand side, 75 minus 3 is 72. Now, I want to isolate this x squared. I have a 2x squared here. So I could have just an x squared here if I divide this side or really both sides by 2. Anything I do to one side, I have to do to the other side if I want to maintain the equality. So the left side, just becomes x squared. And the right hand side is 72 divided by 2 is 36. So we're left with x squared is equal to 36. And then to solve for x, we can take the positive, the plus or minus square root of both sides. So we could say the plus or-- let me write it this way-- If we take the square root of both sides, we would get x is equal to the plus or minus square root of 36, which is equal to plus or minus 6. Let me just write that on another line. So x is equal to plus or minus 6. And remember here, if something squared is equal to 36, that something could be the negative version or the positive version. It could be the principal root or it could be the negative root. Both negative 6 squared is 36 and positive 6 squared is 36, so both of these work. And you could put them back into the original equation to verify it. Let's do that. If you say 2 times 6 squared plus 3, that's 2 times 36, which is 72 plus 3 is 75. So that works. If you put negative 6 in there, you're going to get the exact same result. Because negative 6 squared is also 36. 2 times 36 is 72 plus 3 is 75.