Question

Find the number to add to x^2+14
to make it a perfect square trinomial. Write that trinomial as the square of a binomial.

Answer 1

To convert the expression \(x^2 + 14\) into a perfect square trinomial, we need to determine the appropriate constant to add.

A perfect square trinomial takes the form \((x + a)^2\) which expands to \(x^2 + 2ax + a^2\).

From your given expression \(x^2 + 14\), we can identify that it currently lacks the linear term \(2ax\).

To make this a perfect square trinomial, we first need to establish our goal, which is to express \(x^2 + 14 + k\) in the form of a perfect square. Adding a term \(k\) will create a trinomial which will be of the form \(x^2 + 14 + k = (x + b)^2\) for some \(b\).

1. **First, rewrite the expression**:
\[
x^2 + 14 + k
\]

2. **Set the right part equal to the squared binomial**:
\[
x^2 + 14 + k = (x + b)^2 = x^2 + 2bx + b^2
\]

3. **Equating the constant terms**:
To achieve this, we match the constant terms:
\[
14 + k = b^2
\]
Hence, the equation we need is:
\[
k = b^2 - 14
\]

4. **Choosing a value of \(b\)**:
To make calculations simple, let's choose \(b = 4\):
\[
k = 4^2 - 14 = 16 - 14 = 2
\]

5. **Adding this back to our original expression**:
Therefore, we need to add 2 to get:
\[
x^2 + 14 + 2 = x^2 + 16
\]

6. **Expressing as a perfect square**:
The expression \(x^2 + 16\) can now be expressed as:
\[
x^2 + 16 = x^2 + 2(4)x + 4^2 = (x + 4)^2
\]

Thus, the number you need to add to \(x^2 + 14\) to make it a perfect square trinomial is \(2\), and the trinomial can be written as the square of a binomial:

\[
(x + 4)^2
\]