For questions $$ 1-5 $$, factor the quadratic function using an algebraic method. For questions $$ \begin{array}{lll}1-2, ext { draw a diagram of algebra tiles that could be used to model the factored function. } \ \begin{array}{lll} ext { 1. } f(x)=x^{2}+8 x+16 & ext { 2. } g(x)=4 x^{2}-25 & ext { 3. } h(x)=9 x^{2}+30 x+25 \ ext { 4. } p(x)=\frac{1}{4} x^{2}+7 x+49 & ext { 5. } q(x)=\frac{1}{16} x^{2}-9 & \end{array}\end{array} $$

Question

For questions $$ 1-5 $$, factor the quadratic function using an algebraic method. For questions $$ \begin{array}{lll}1-2, 	ext { draw a diagram of algebra tiles that could be used to model the factored function. } \ \begin{array}{lll}	ext { 1. } f(x)=x^{2}+8 x+16 & 	ext { 2. } g(x)=4 x^{2}-25 & 	ext { 3. } h(x)=9 x^{2}+30 x+25 \ 	ext { 4. } p(x)=\frac{1}{4} x^{2}+7 x+49 & 	ext { 5. } q(x)=\frac{1}{16} x^{2}-9 & \end{array}\end{array}> $$

UpStudy AI · Accepted Answer

**1. Factor $$f(x)=x^2+8x+16$$**

We look for two numbers that add to $$8$$ and multiply to $$16$$. Since
$$
4+4=8 \quad\text{and}\quad 4\cdot4=16,
$$
we factor
$$
x^2+8x+16=(x+4)(x+4)=(x+4)^2.
$$

*Algebra Tiles Diagram for $$f(x)$$:*

To model $$(x+4)^2$$ with algebra tiles, one represents a larger square whose side is composed of an $$x$$-tile and $$4$$ unit tiles. For example, arrange the tiles as follows:

- Create a square that is $$2$$ tiles by $$2$$ tiles in the sense of algebra tile blocks.
- The top edge is represented by an $$x$$-tile adjoining $$4$$ unit tiles (divided equally between the top and side of the square).  
- This yields one $$x^2$$ tile (the large square), a total of $$8$$ $$x$$-tiles (distributed as $$4$$ on the top edge and $$4$$ on the side edge), and $$16$$ unit tiles filling the remaining area.

A simplified sketch might look like:

```
+-----------+-----------+ ... +-----------+
|    x      |   1-unit  | ... |   1-unit  |
+-----------+-----------+ ... +-----------+
| 1-unit   | 1-unit   | ... | 1-unit    |
+-----------+-----------+ ... +-----------+
         ... (total of 4 unit tiles along side)
```

This diagram represents forming a large square with side length corresponding to $$x+4$$.

---

**2. Factor $$g(x)=4x^2-25$$**

Notice that
$$
4x^2-25=(2x)^2-5^2,
$$
which is a difference of two squares. Thus, we factor
$$
4x^2-25=(2x-5)(2x+5).
$$

*Algebra Tiles Diagram for $$g(x)$$:*

For a difference of squares, one can imagine a large square representing $$4x^2=(2x)^2$$ from which a smaller square of area $$25=5^2$$ is removed. If you form a rectangle with dimensions corresponding to the binomials, the diagram will have a partition so that one dimension is labeled $$2x-5$$ and the other $$2x+5$$. A rough sketch could be:

```
         _____________
        |      |      |
        |      |      |
        | 2x-5 |      |
        |      | 2x+5|
        |______|______|
```

Here, the overall area is the difference of the two squares, which corresponds to the factorization.

---

**3. Factor $$h(x)=9x^2+30x+25$$**

Examine the trinomial:
- The coefficient $$9$$ is a perfect square $$(3x)^2$$.
- The constant $$25$$ is a perfect square $$5^2$$.
- And $$2\cdot3x\cdot5=30x$$.

Thus, we have a perfect square:
$$
9x^2+30x+25=(3x+5)^2.
$$

---

**4. Factor $$p(x)=\frac{1}{4}x^2+7x+49$$**

To eliminate the fraction, multiply the entire function by $$4$$:
$$
4p(x)=x^2+28x+196.
$$
Now, note that
$$
x^2+28x+196=(x+14)^2,
$$
since $$14+14=28$$ and $$14^2=196$$. Therefore,
$$
4p(x)=(x+14)^2\quad\Longrightarrow\quad p(x)=\frac{1}{4}(x+14)^2.
$$

---

**5. Factor $$q(x)=\frac{1}{16}x^2-9$$**

Write the expression as a difference of squares:
$$
q(x)=\frac{1}{16}x^2-9=\left(\frac{x}{4}\right)^2-3^2.
$$
This factors as
$$
\left(\frac{x}{4}-3\right)\left(\frac{x}{4}+3\right).
$$
Alternatively, factor out $$\frac{1}{16}$$:
$$
q(x)=\frac{1}{16}\left(x^2-144\right)=\frac{1}{16}(x-12)(x+12).
$$

**1. $$f(x)=x^2+8x+16 = (x+4)^2$$ 2. $$g(x)=4x^2-25 = (2x-5)(2x+5)$$ 3. $$h(x)=9x^2+30x+25 = (3x+5)^2$$ 4. $$p(x)=\frac{1}{4}x^2+7x+49 = \frac{1}{4}(x+14)^2$$ 5. $$q(x)=\frac{1}{16}x^2-9 = \frac{1}{16}(x-12)(x+12)$$

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