https://file.aiquickdraw.com/mathbot/file/1726365304242zeentflh.jpg

The image contains two mathematical expressions that need to be simplified using notable identities.

Here are the expressions:

1. \( x + x^2 + \frac{1}{4} \)
2. \( \frac{49}{169} a^6 - 16 b^4 \)

### 1. Simplification of \( x + x^2 + \frac{1}{4} \):
We can rewrite the expression as follows:

\[ x + x^2 + \frac{1}{4} = x^2 + x + \frac{1}{4} \]

Now, notice that this can be factored by completing the square:

\[ x^2 + x + \frac{1}{4} = \left(x + \frac{1}{2}\right)^2 \]

Thus, the compact form is:

\[
x + x^2 + \frac{1}{4} = \left(x + \frac{1}{2}\right)^2
\]

### 2. Simplification of \( \frac{49}{169} a^6 - 16 b^4 \):
This is a difference of squares. Recall that:

\[
A^2 - B^2 = (A - B)(A + B)
\]

We need to express each term as a square. Notice that:

\[
\frac{49}{169} a^6 = \left(\frac{7}{13} a^3\right)^2
\quad \text{and} \quad
16 b^4 = (4 b^2)^2
\]

Thus, applying the difference of squares:

\[
\frac{49}{169} a^6 - 16 b^4 = \left(\frac{7}{13} a^3 - 4 b^2\right) \left(\frac{7}{13} a^3 + 4 b^2\right)
\]

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Would you like further details on any part of the process, or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you complete the square in quadratic expressions?
2. What are some other common notable identities used in algebra?
3. Can the difference of cubes be factored similarly to the difference of squares?
4. How does factoring help in solving equations?
5. What are the conditions for using the difference of squares identity?

**Tip:** When simplifying expressions, always look for patterns like squares, cubes, or other notable identities that can reduce the expression to a more manageable form.

Rewrite the following expressions in a more compact form:
1) x + x^2 + 1/4 
2) 49/169 a^6 - 16 b^4

Simplify the expressions 'x + x^2 + 1/4' and '49/169 a^6 - 16 b^4' using factoring techniques like completing the square and the difference of squares. Ideal for high school students studying algebra.

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