By completing the square, solve the equation one plus 𝑥 equals 𝑥 squared.
So when I get a problem like this and I’m asked to solve a quadratic, which is what this is, I like to get it into the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero first. So I’ve got my 𝑥 squared term first, then my 𝑥 term, then I’ve got the constant, and then this is equal to zero.
And in order to do this, what I’m gonna do is subtract 𝑥 from each side of the equation. I’m gonna do this because I want to keep the 𝑥 squared term positive. And when I do this, I get one is equal to 𝑥 squared minus 𝑥. And then, next, subtract one from each side of the equation. When I do this, I get zero is equal to 𝑥 squared minus 𝑥 minus one, which I can then rewrite with a zero on the right-hand side. So we’ve got 𝑥 squared minus 𝑥 minus one equals zero. Okay, great. We’ve got it in the way that I wanted.
So now what we need to do is look at completing the square, because the question asked us to use completing the square to solve the equation. So to enable us to complete the square, we’ve got this general rule. That’s if we have 𝑥 squared plus 𝑏𝑥 — so we have it in the form 𝑥 squared plus 𝑏𝑥, where 𝑏 is the coefficient of 𝑥 and 𝑥 squared is a single 𝑥 squared term — this is equal to 𝑥 plus 𝑏 over two — so the coefficient of 𝑥 over two; then this is all squared — minus 𝑏 over two squared, where again 𝑏 is the coefficient of 𝑥.
So now if we take a look at our equation, we can see that, in our equation, the coefficient of 𝑥 is negative one. So therefore, our 𝑏-value is gonna be equal to negative one. So what we can do is use this to rearrange our equation into the form that involves completing the square.
When I do this, I’m gonna get 𝑥 minus a half all squared. I get minus a half because if we have half of negative one, we’re gonna get negative a half. Then minus negative a half all squared minus one equals zero. And then when we simplify, we get 𝑥 minus a half, and this is all squared, minus five over four or five-quarters equals zero. And we got the negative five over four or negative five-quarters because we had negative a half squared, which is a quarter, and then we had a negative sign in front of that, so we get negative a quarter and then minus one, where one is the same as four-quarters. So we’re gonna get negative five-quarters.
So now we’ve done the first stage and we’ve completed the square. What we want to do is solve the equation. And the first step to solve an equation is to add five-quarters or five over four to each side of the equation. And when we do that, we get 𝑥 minus a half all squared is equal to five over four. Then the next stage is to take the square root of each side of the equation. That’s cause we’ve got 𝑥 minus a half all squared. So the inverse is to square-root. And when we do that, we get 𝑥 minus a half is equal to positive or negative root five over four.
It’s very important to remember both positive and negative answers could be used here. Then the next step is to add a half to each side of the equation, so we get 𝑥 is equal to positive or negative root five over four plus a half. And now we can simplify a step further, cause we can use a rule we know. And that rule is that root 𝑎 over 𝑏 is equal to root 𝑎 over root 𝑏.
So therefore, we can use it on ours because we can say that root five over four is gonna be equal to root five over root four. So therefore, we’re gonna get 𝑥 is equal to, and I’ve put the half in front here so we’ve got a half plus or minus then root five over two, and that’s because root four is two. And as we’ve got two fractions here with the same denominator, cause we have a half and root five over two, then we can just add or subtract the numerators. So therefore, we can say that, by completing the square, the solutions to the equation one plus 𝑥 equals 𝑥 squared are 𝑥 equals one plus root five over two or one minus root five over two.