Answer to College algebra problem. Here is a photo

Luis Rojas

Answered question

2022-07-03

alenahelenash

Expert2023-05-29Added 556 answers

To determine the constant that should be added to the binomial

x^{2} + \frac{1}{6} x

in order to make it a perfect square trinomial, we need to consider the square of a binomial pattern.
The square of the binomial

(a + b)^{2}

can be expanded as follows:

(a + b)^{2} = a^{2} + 2 a b + b^{2}

Comparing this pattern to the given binomial

x^{2} + \frac{1}{6} x

, we can see that

a^{2}

corresponds to

x^{2}

2 a b

corresponds to

\frac{1}{6} x

, and

b^{2}

corresponds to the unknown constant we need to find.
From the comparison, we can deduce that

2 a b = \frac{1}{6} x

. In this case,

a = x

and

b

is the unknown constant we're looking for.
To find

b

, we can rewrite the equation as:

2 a b = \frac{1}{6} x

Substituting

a = x

2 x \cdot b = \frac{1}{6} x

Now, we can solve for

b

b = \frac{\frac{1}{6} x}{2 x} = \frac{1}{12}

Therefore, the constant that needs to be added to the binomial

x^{2} + \frac{1}{6} x

to make it a perfect square trinomial is

\frac{1}{12}

.
Now, let's write and factor the trinomial.
The original binomial is

x^{2} + \frac{1}{6} x

. To make it a perfect square trinomial, we add

{(\frac{1}{12})}^{2}

to it:

x^{2} + \frac{1}{6} x + {(\frac{1}{12})}^{2} = x^{2} + \frac{1}{6} x + \frac{1}{144}

Now, we can factor the trinomial:

x^{2} + \frac{1}{6} x + \frac{1}{144} = {(x + \frac{1}{12})}^{2}

And that's the factored form of the trinomial.

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