26. modeling real life a dentist’s office and parking lot are on a rectangular piece of land. the area (in square meters) of the land is represented by (x^2 + x - 30).
image of a rectangle with labels (x) m, ((x - 8)) m, ((x + 6)) m
a. write a binomial that represents the width of the land.
b. find the perimeter of the land when the length of the dentist’s office is 20 meters.
28. (n^2 - 9n + 18 = 0)
30. (x^2 + 5x - 14 = 0)
32. (t^2 + 15t = -36)
34. (y^2 - 2y - 8 = 7)
36. (b^2 + 5 = 8b - 10)
38. modeling real life you are decorating a parade float for a cinco de mayo parade. the area of the entire parade float is 192 square feet. what is the area covered by paper flowers?
image of a rectangle with a smaller rectangle, labels 9 ft, ((x + 12)) ft, 5 ft, (x) ft
cinco de mayo commemorates the mexican army’s 1862 victory at the battle of puebla.

Question

26. modeling real life a dentist’s office and parking lot are on a rectangular piece of land. the area (in square meters) of the land is represented by (x^2 + x - 30).
image of a rectangle with labels (x) m, ((x - 8)) m, ((x + 6)) m
 a. write a binomial that represents the width of the land.
 b. find the perimeter of the land when the length of the dentist’s office is 20 meters.
28. (n^2 - 9n + 18 = 0)
30. (x^2 + 5x - 14 = 0)
32. (t^2 + 15t = -36)
34. (y^2 - 2y - 8 = 7)
36. (b^2 + 5 = 8b - 10)
38. modeling real life you are decorating a parade float for a cinco de mayo parade. the area of the entire parade float is 192 square feet. what is the area covered by paper flowers?
image of a rectangle with a smaller rectangle, labels 9 ft, ((x + 12)) ft, 5 ft, (x) ft
cinco de mayo commemorates the mexican army’s 1862 victory at the battle of puebla.

Accepted Answer

### Problem 26
# Explanation:
## Part a
## Step1: Factor the area polynomial
Factor $x^2 + x - 30$. We need two numbers that multiply to $-30$ and add to $1$, which are $6$ and $-5$.
$$x^2 + x - 30 = (x + 6)(x - 5)$$
## Step2: Identify width binomial
The land's length is given as $(x + 6)$ m, so the width is the other factor.
# Answer (Part a):
$\boldsymbol{x - 5}$

## Part b
## Step1: Relate office length to $x$
The dentist's office length is $x$ m, set equal to 20.
$$x = 20$$
## Step2: Calculate width value
Substitute $x=20$ into width binomial $x - 5$.
$$	ext{Width} = 20 - 5 = 15$$
## Step3: Calculate perimeter of rectangle
Use perimeter formula $P = 2(	ext{length} + 	ext{width})$, length is $x+6=26$.
$$P = 2(26 + 15) = 2(41) = 82$$
# Answer (Part b):
$\boldsymbol{82}$ meters

---

### Problem 28
# Explanation:
## Step1: Factor the quadratic equation
Find two numbers that multiply to $18$ and add to $-9$: $-3$ and $-6$.
$$n^2 - 9n + 18 = (n - 3)(n - 6) = 0$$
## Step2: Solve for $n$
Set each factor equal to 0 and solve.
$$n - 3 = 0 \implies n=3; \quad n - 6 = 0 \implies n=6$$
# Answer:
$\boldsymbol{n=3}$ and $\boldsymbol{n=6}$

---

### Problem 30
# Explanation:
## Step1: Factor the quadratic equation
Find two numbers that multiply to $-14$ and add to $5$: $7$ and $-2$.
$$x^2 + 5x - 14 = (x + 7)(x - 2) = 0$$
## Step2: Solve for $x$
Set each factor equal to 0 and solve.
$$x + 7 = 0 \implies x=-7; \quad x - 2 = 0 \implies x=2$$
# Answer:
$\boldsymbol{x=-7}$ and $\boldsymbol{x=2}$

---

### Problem 32
# Explanation:
## Step1: Rewrite in standard form
Add 36 to both sides to set equation to 0.
$$t^2 + 15t + 36 = 0$$
## Step2: Factor the quadratic equation
Find two numbers that multiply to $36$ and add to $15$: $12$ and $3$.
$$(t + 12)(t + 3) = 0$$
## Step3: Solve for $t$
Set each factor equal to 0 and solve.
$$t + 12 = 0 \implies t=-12; \quad t + 3 = 0 \implies t=-3$$
# Answer:
$\boldsymbol{t=-12}$ and $\boldsymbol{t=-3}$

---

### Problem 34
# Explanation:
## Step1: Rewrite in standard form
Subtract 7 from both sides to set equation to 0.
$$y^2 - 2y - 15 = 0$$
## Step2: Factor the quadratic equation
Find two numbers that multiply to $-15$ and add to $-2$: $-5$ and $3$.
$$(y - 5)(y + 3) = 0$$
## Step3: Solve for $y$
Set each factor equal to 0 and solve.
$$y - 5 = 0 \implies y=5; \quad y + 3 = 0 \implies y=-3$$
# Answer:
$\boldsymbol{y=5}$ and $\boldsymbol{y=-3}$

---

### Problem 36
# Explanation:
## Step1: Rewrite in standard form
Subtract $8b$ and add 10 to both sides.
$$b^2 - 8b + 15 = 0$$
## Step2: Factor the quadratic equation
Find two numbers that multiply to $15$ and add to $-8$: $-3$ and $-5$.
$$(b - 3)(b - 5) = 0$$
## Step3: Solve for $b$
Set each factor equal to 0 and solve.
$$b - 3 = 0 \implies b=3; \quad b - 5 = 0 \implies b=5$$
# Answer:
$\boldsymbol{b=3}$ and $\boldsymbol{b=5}$

---

### Problem 38
# Explanation:
## Step1: Set up total area equation
Total length is $9 + (x + 12) = x + 21$, total width is $5 + x$. Area is 192.
$$(x + 21)(x + 5) = 192$$
## Step2: Expand and simplify equation
Expand left side and subtract 192 to set to 0.
$$x^2 + 26x + 105 - 192 = x^2 + 26x - 87 = 0$$
## Step3: Factor the quadratic equation
Find two numbers that multiply to $-87$ and add to $26$: $29$ and $-3$.
$$(x + 29)(x - 3) = 0$$
## Step4: Solve for positive $x$
Discard negative solution, so $x=3$.
## Step5: Calculate flower area
Flower area is $(x + 12)x$, substitute $x=3$.
$$(3 + 12)(3) = 15 	imes 3 = 45$$
# Answer:
$\boldsymbol{45}$ square feet

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 26

Part a

Step1: Factor the area polynomial

Step2: Identify width binomial

Step1: Factor the quadratic equation

Step2: Solve for $n$

Step1: Factor the quadratic equation

Step2: Solve for $x$

Answer:

Part b

Step1: Relate office length to $x$

Step2: Calculate width value

Step3: Calculate perimeter of rectangle