Algebra 2 3.5-3.8

Question

Fill in the blank with a number to make the expression a perfect square.

W^2- 16w+

Answer 1

You want to find a number x that makes w² - 16 w + x be a perfect square.

If w² - 16 w + x really was a perfect square, you'd be able to factor it into two identical factors like this:

w² - 16 w + x = (w + something)(w + something)

Let's find what this "something" is. Multiplying out,

w² - 16 w + x = (w + something)(w + something) = w² + w·something + w·something + something²

= w² + 2w·something + something²

So now we know that

w² - 16 w + x = w² + 2w·something + something²

The middle terms in the above equations must be equal.

Therefore -16w = 2w·something.

Therefore -16 = 2·something.

Therefore something = -8.

So now we know that

w² - 16 w + x = (w + something)(w + something) = (w - 8)(w - 8) = w -16w + 64.

The right hand terms in the above equations must be equal.

Therefore x = 64.

Therefore w² - 16w + 64 is a perfect square.

Therefore the number that fills the blank in the original problem is 64.

Answer 2

To complete the square for a quadratic of the form x² +/- bx + ___ , follow these steps:

1. Divide b by 2.
2. Square the value you get in Step 1.

Here are some examples:

If the original problem had been

x² + 10x + ___, you would divide 10 by 2 which gives us 5. Then we square 5 and get 25.

Therefore we would need to add 25 to make this a perfect square trinomial.

One more example:

x² - 8x + ____, you would divide -8 by 2 which gives -4. Then we square -4 and get 16 (remember that when we square a negative we still get a positive. Therefore we would need to add 16 to get a perfect square trinomial.

Hopefully that helps and you can try that same strategy with your specific problem!