Question Video: Solving Quadratic Equations by Completing the Square Mathematics

Video Transcript

Solve the equation 𝑥 squared minus 14 𝑥 plus 38 is equal to zero by completing the square.

This question has asked us to solve the equation by completing the square. So what I’m gonna do is I’m going to recap that first. So the general rule if we have our expression in the form 𝑥 squared plus 𝑎𝑥 is that this is equal to 𝑥 plus 𝑎 over two all squared — and that’s because we’ve actually halved the coefficient of 𝑥 — and then minus 𝑎 over two — again halving the coefficient of 𝑥 — and that is all squared.

So if we look back to our equation, we can actually see that it’s the first two terms that are actually gonna apply to completing the square rule too. And these two terms are 𝑥 squared minus 14𝑥. So if we actually apply this and complete the square of our first two terms, we’re gonna get 𝑥 plus and then we’ve got negative 14 over two because a coefficient of our 𝑥 is negative 14 and that’s all squared and then minus again negative 14 over two all squared. Then we still have plus 38 is equal to zero.

Okay, we can tidy this up. So we get 𝑥 minus seven all squared and it’s 𝑥 minus seven because we had 𝑥 plus and then negative 14 over two. Well, negative 14 over two is negative seven. So if you add a negative, it’s the same as just subtracting it. And then minus 49. And we get that again because we had negative seven all squared. If you square a negative, we get a positive. And then add 38 is equal to zero. So then if we simplify, we get 𝑥 minus seven all squared minus 11 is equal to zero.

And then our next stage is to actually add 11 to each side of the equation, which gives us 𝑥 minus seven all squared is equal to 11. And then if we take the square root of each side, we get 𝑥 minus seven is equal to plus or minus the square root of 11.

And now into our final stage, which is actually going to be let’s add seven to each side of the equation. So then, we’re left with 𝑥 is equal to seven plus or minus root 11. So therefore, the solutions to the equation 𝑥 squared minus 14𝑥 plus 38 equals zero are 𝑥 is equal to seven plus root 11 or 𝑥 is equal to seven minus root 11. And we found these by completing the square.