Video Transcript
By completing the square, solve the equation one plus ๐ฅ equals ๐ฅ squared.
So when I get a problem like this and Iโm asked to solve a quadratic, which is what this is, I like to get it into the form ๐๐ฅ squared plus ๐๐ฅ plus ๐ equals zero first. So Iโve got my ๐ฅ squared term first, then my ๐ฅ term, then Iโve got the constant, and then this is equal to zero.
And in order to do this, what Iโm gonna do is subtract ๐ฅ from each side of the equation. Iโm gonna do this because I want to keep the ๐ฅ squared term positive. And when I do this, I get one is equal to ๐ฅ squared minus ๐ฅ. And then, next, subtract one from each side of the equation. When I do this, I get zero is equal to ๐ฅ squared minus ๐ฅ minus one, which I can then rewrite with a zero on the right-hand side. So weโve got ๐ฅ squared minus ๐ฅ minus one equals zero. Okay, great. Weโve 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, weโve got this general rule. Thatโs if we have ๐ฅ squared plus ๐๐ฅ โ so we have it in the form ๐ฅ squared plus ๐๐ฅ, where ๐ is the coefficient of ๐ฅ and ๐ฅ squared is a single ๐ฅ squared term โ this is equal to ๐ฅ plus ๐ over two โ so the coefficient of ๐ฅ over two; then this is all squared โ minus ๐ over two squared, where again ๐ is the coefficient of ๐ฅ.
So now if we take a look at our equation, we can see that, in our equation, the coefficient of ๐ฅ is negative one. So therefore, our ๐-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, Iโm gonna get ๐ฅ minus a half all squared. I get minus a half because if we have half of negative one, weโre gonna get negative a half. Then minus negative a half all squared minus one equals zero. And then when we simplify, we get ๐ฅ 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 weโre gonna get negative five-quarters.
So now weโve done the first stage and weโve 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 ๐ฅ 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. Thatโs cause weโve got ๐ฅ minus a half all squared. So the inverse is to square-root. And when we do that, we get ๐ฅ minus a half is equal to positive or negative root five over four.
Itโs 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 ๐ฅ 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 ๐ over ๐ is equal to root ๐ over root ๐.
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, weโre gonna get ๐ฅ is equal to, and Iโve put the half in front here so weโve got a half plus or minus then root five over two, and thatโs because root four is two. And as weโve 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 ๐ฅ equals ๐ฅ squared are ๐ฅ equals one plus root five over two or one minus root five over two.
