Completing the Square - Finding a value for k in x2+4x+k

[markl77]

I came across this question:
Determine the value of k that makes each expression a perfect square trinomial.
x²+4x+k

Here is the work I did:
1(x²+4x)+k
(x²+4x+4-4)+k
=1(x+2)²+k
I am stuck here and have no idea what to do.

 

[Harry_the_cat]

I came across this question:
Determine the value of k that makes each expression a perfect square trinomial.
x²+4x+k

Here is the work I did:
1(x²+4x)+k
(x²+4x+4-4)+k
=1(x+2)²+k
I am stuck here and have no idea what to do.

A perfect square trinomial can be expressed as \(\displaystyle (x+n)^2\)

Now \(\displaystyle (x+n)^2 = x^2 + 2nx +n^2\)

ie if the first term is \(\displaystyle x^2\), then the coefficient of \(\displaystyle x\) is \(\displaystyle 2n\) and the constant term at the end is \(\displaystyle n^2\).

In your case, the coefficient of \(\displaystyle x\) is \(\displaystyle 4\), so \(\displaystyle 2n=4\), ie \(\displaystyle n=2\).

Therefore the constant term \(\displaystyle k=2^2=4\).

Check: \(\displaystyle x^2 + 4x +4=(x+2)^2\) which is a perfect square trinomial.

 

[Harry_the_cat]

I came across this question:
Determine the value of k that makes each expression a perfect square trinomial.
x²+4x+k

Here is the work I did:
1(x²+4x)+k
(x²+4x+4-4)+k .... completing the square
=(x²+4x+4)-4+k

=1(x+2)² - 4 +k

To be a perfect square, -4+k=0, so k=4

I am stuck here and have no idea what to do.

Alternatively, using your approach (after correcting your error), see above.
(This is the long way around though.)

 

[ksdhart2]

You're very very close, and everything you've done so far is absolutely correct. I think you're just overthinking the process a bit. As you know, when you complete the square you take a polynomial that looks like ax² + bx and turn it into one that looks like a(x+b/2)² + k. The given polynomial has an a of 1, so that makes the process that much easier. Let's temporarily forget about the k term and imagine you were asked to complete the square of x² + 4x. You'd do exactly the same steps you did here, arriving at x² + 4x + 4 - 4. Now, we'll group up the parts that are a perfect square, leaving (x² + 4x + 4) - 4, = (x + 2)² - 4.

However, the exercise didn't ask you to complete the square, per se. It asked you to find a value of k that makes x^2 + 4x + k a perfect square. The perfect square we found above is (x +2)². What value of k would you use to make the given expression into that perfect square?

 

[markl77]

You're very very close, and everything you've done so far is absolutely correct. I think you're just overthinking the process a bit. As you know, when you complete the square you take a polynomial that looks like ax² + bx and turn it into one that looks like a(x+b/2)² + k. The given polynomial has an a of 1, so that makes the process that much easier. Let's temporarily forget about the k term and imagine you were asked to complete the square of x² + 4x. You'd do exactly the same steps you did here, arriving at x² + 4x + 4 - 4. Now, we'll group up the parts that are a perfect square, leaving (x² + 4x + 4) - 4, = (x + 2)² - 4.

However, the exercise didn't ask you to complete the square, per se. It asked you to find a value of k that makes x^2 + 4x + k a perfect square. The perfect square we found above is (x +2)². What value of k would you use to make the given expression into that perfect square?

Thanks! I guess I just didn't really read the question. (It was sectioned under "Completing the Square", so I assumed that I had to do that.) This helps a lot though.