Help needed please

I don’t know where to start with these questions I was given as homework and would really appreciate help. Questions below

They are all effectively the same question, so what you need to know is how to complete the square. Here's an example of how it works (I've made up the numbers without looking in detail at your questions, so I hope they are not the same as any of yours).

Suppose we've got the quadratic x^2 - 8x + 3. Ignore the + 3 for the moment - it is still there, but we are just going to leave it alone for now.

You should be able to see that x^2 - 8x = (x - 4)^2 - 4^2. The way we have done this is to write a bracket of the form (x - k)^2 so that when it is expanded out, we get back the same x^2 and x term, and the only way this will work is if k is half of the original number of x's we had. We started off with -8x, so k = -4, half of -8. But there is now a problem - if we multiply out (x - 4)^2, we do get x^2 and -8x, but we also get 4^2 as well. To make sure that things balance out, we also have to subtract this 4^2, and so x^2 - 8x = (x - 4)^2 - 4^2.

Now we need to remember the + 3 that was there all the time. A full soultion would look like this;

x^2 - 8x +3 = (x - 4)^2 - 4^2 +3 = (x - 4)^2 - 16 + 3 = (x - 4)^2 -13.

And that is the form that you want. Now try this with your questions.

There's a nice picture that explains why this is called "completing the square" - have a look at the Wikipedia page about this.

They are all effectively the same question, so what you need to know is how to complete the square. Here's an example of how it works (I've made up the numbers without looking in detail at your questions, so I hope they are not the same as any of yours).

Suppose we've got the quadratic x^2 - 8x + 3. Ignore the + 3 for the moment - it is still there, but we are just going to leave it alone for now.

You should be able to see that x^2 - 8x = (x - 4)^2 - 4^2. The way we have done this is to write a bracket of the form (x - k)^2 so that when it is expanded out, we get back the same x^2 and x term, and the only way this will work is if k is half of the original number of x's we had. We started off with -8x, so k = -4, half of -8. But there is now a problem - if we multiply out (x - 4)^2, we do get x^2 and -8x, but we also get 4^2 as well. To make sure that things balance out, we also have to subtract this 4^2, and so x^2 - 8x = (x - 4)^2 - 4^2.

Now we need to remember the + 3 that was there all the time. A full soultion would look like this;

x^2 - 8x +3 = (x - 4)^2 - 4^2 +3 = (x - 4)^2 - 16 + 3 = (x - 4)^2 -13.

And that is the form that you want. Now try this with your questions.

There's a nice picture that explains why this is called "completing the square" - have a look at the Wikipedia page about this.

I’ve got this
What about finding the least value of x^2+8x+2?

This is one of the main reasons for completing the square. It is not easy to tell the least value of the quadratic (that is, the smallest value you get out when you put in different value of x) when it is written in its original form. But now that you have completed the square, look at the two parts you have got. One is (x + 4)^2, and the other is -14. Now, -14 is -14, different values of x can't change that. But what about (x + 4)^2? What is the smallest value this can have as you let x take different values? If you can see this, you will have an answer to the question.

I don’t really see it but I’m thinking -2?

Can I ask why you think that? What about just looking at the (x + 4)^2 part - what is the smallest value that part can have if x is allowed to be anything? Resist the temptation to expand the bracket (the whole point was to get away from looking at the quadratic in that way).

Is the answer just -14?

It is, and I hope that you can see that this is really easy to see when you have completed the square, but quite hard to see if you haven't. And that is one of the main reasons for bothering to complete the square in the first place.

A bit of general advice - you'll probably get a better response to your questions with clearer thread titles. For example, you could have called this one "Completing the square help" or similar. People will know what you want help with and will be more likely to look at your threads.