Solve the equation negative 𝑥 squared plus seven 𝑥 plus one equals zero.
We can start by copying down the equation exactly as it was listed in the problem. And then I noticed this negative leading coefficient.
Since I’m going to have to do some factoring or may be completing the square, I want to make sure that my leading coefficient is positive, and I can do this by moving it to the other side of the equation. I add 𝑥 squared to both sides.
But I don’t only want to move my leading coefficient; I wanna keep the whole equation on the same side. This means that I’ll subtract seven from both sides of the equation as well. I’ll also subtract one from both sides of the equation. Essentially, we’ve just flipped the problem. We now have zero equals 𝑥 squared minus seven 𝑥 minus one.
I’m starting to think about how to factor and solve this equation, and I looked to the third term. If I wanted to factor the problem like this, I would need to take two factors of the third term, something like this, 𝑥 minus one 𝑥 plus one. The problem is, we have this third term of negative seven, so this kind of factoring won’t help us solve this problem. To solve this problem, we’ll have to use a strategy called completing the square.
The first step here would be to move the whole number one back to the other side of the equation. We’re left with one equals 𝑥 squared minus seven 𝑥. Completing the square works by taking the coefficient of the middle term, the b here. We then use whatever that b value is, and we divide it by two and then we square it.
After we do that, we add that value to both sides of the equation. In our case, the b equals negative seven, so we’ll need to add negative seven over two squared to both sides of our equation. That would be negative seven squared over two squared, which simplifies to 49 over four. We need to add 49 over four to both sides of our equation.
Notice that, instead of saying 49 over four plus one, I said 49 over four plus four over four. Four over four is equal to one, and we can go ahead and add these terms together if we give them a common denominator. The rest we just copy down. 49 fourths plus four-fourths equals 53 fourths.
Now we need to factor 𝑥 squared minus seven 𝑥 plus 49 over four. And if you didn’t immediately recognize the pattern, that’s okay, because completing the square tells us that the factor of this problem will be equal to 𝑥 plus b over two squared. We add 𝑥 plus; remember our b was negative seven over two squared, but we still haven’t solved the equation.
Solving the equation means we know what 𝑥 equals. Now we’ll need to get 𝑥 completely by itself. To get rid of this square, we’ll need to take the square root of both sides of our equation. Now we’re left with the square root of 53 over the square root of four equals 𝑥 plus negative seven over two.
From this line, we can take the square root of four, which equals two. We’ll keep the square root of 53 and say that we need the positive square root of 53 and the negative square root of 53.
To get 𝑥 by itself, we’ll need to add seven halves to both sides of the equation. On the right side, negative seven halves and positive seven halves cancel out. On the left side, we have seven halves plus or minus the square root of 53 over two. That plus or minus there tells us that there will be two solutions here.
The first solution will be seven plus the square root of 53 over two and the second solution would be seven minus the square root of 53 over two. These are the two solutions of the equation negative 𝑥 squared plus seven 𝑥 plus one.