Finding the square with a fraction in the expression

[Amaz1ng]

Homework Statement

$$x^2 + x + \frac{1}{4}$$

Homework Equations

This should be written in the form:

$$(x+y)^2$$

Just to add a bit more info, the exercise is to "Write each of the following as the square of a binomial expression". So basically the book teachs to take the rook of x^2 and 1/2, multiply those together, then multiply by 2. If that is equal to the middle term, then you can write:

$$(\sqrt{x^2} + \sqrt{1/4})^2$$

The Attempt at a Solution

My answer is that this can't be written as a square...which is what the textbook is asking to do. Anyway, I don't think this can be written as a square

 

[Karlx]

Hi Amaz1ng.

You are asked to find if there is a value a so you can write

$$x^{2}+x+1/4=(x+a)^{2}$$

 

[SteamKing]

If I have a number a, and I know a^2 = 1/4, what is a?

 

[Mark44]

Hi Amaz1ng.

You are asked to find if there is a value a so you can write

$$x^{2}+x+1/4=(x+a)^{2}$$

I don't believe there is such a value of a. You can, however write x² + x + 1/4 as (x + a)² + b.

This technique is called completing the square. Your textbook should have numerous examples of how to do this.

 

[Amaz1ng]

Just to add a bit more info, the exercise is to "Write each of the following as the square of a binomial expression".

 

[eumyang]

Hi Amaz1ng.

You are asked to find if there is a value a so you can write

$$x^{2}+x+1/4=(x+a)^{2}$$

I don't believe there is such a value of a.

Yes there is.

Using the perfect square trinomial pattern
$$a^2 + 2ab + b^2 = (a + b)^2$$
equate
$$x^{2} + x + \frac{1}{4}$$
with the left side to find a and b. If x corresponds to a, what corresponds to b?

 

[Amaz1ng]

answer in book..

$$(x+\frac{1}{2})^2$$

..it's squared but for some reason the square doesn't show.

 

[SteamKing]

Well, has anyone checked to see if (x+1/2) * (x+1/2) = x^2+x+1/4?
Note: this would imply that a = 1/2 from Post #3 above.

 

[Mark44]

Yes there is.

I don't know why I didn't see that.

Using the perfect square trinomial pattern
$$a^2 + 2ab + b^2 = (a + b)^2$$
equate
$$x^{2} + x + \frac{1}{4}$$
with the left side to find a and b. If x corresponds to a, what corresponds to b?