algebra 2h classwork 9/11/2025
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solve the equation by completing the square.
1) x² - 8x - 13 = 0
a) {8 - √77, 8 + √77}
b) {4 - √13, 4 + √13}
c) {-4 - √29, -4 + √29}
d) {4 - √29, 4 + √29}
solve the problem.
2) an object is propelled vertically upward from the top of a 144 - foot building. the quadratic function s(t)= - 16t² + 176t + 144 models the balls height above the ground, s(t), in feet, t seconds after it was thrown. how many seconds after it take until the object finally hits the ground? round to the nearest tenth of a second if necessary.
a) 2 seconds
b) 0.8 seconds
c) 11.8 seconds
d) 5.5 seconds
3) write an expression for the area of the shaded region and express it in factored form.
a) (y + 10)(y - 10)
b) (y - 10)²
c) (y + 10)²
d) y² + 100
4) claire has received scores of 85, 88, 87, and 80 on her algebra tests. what score must she receive on the fifth test to have an overall test score average of at least 82?
a) 70 or greater
b) 71 or greater
c) 68 or greater
d) 69 or greater
5) you have 200 feet of fencing to enclose a rectangular region. find the dimensions of the rectangle that maximize the enclosed area.
6) a square sheet of paper measures 30 centimeters on each side. what is the length of the diagonal of this paper?
a) 30 cm
b) 30√2 cm
c) 60 cm
d) 180 cm
divide using synthetic division.
7) (5x² - 43x + 56)/(x - 7)
a) - 8x - 7
b) - 5x + 8
c) 5x - 8
d) x - 8
8) (2x⁵ + 2x⁴ - 7x³ + x² - x + 128)÷(x + 3)
a) 2x⁴ - 4x³ + 5x² - 15x + 42 + 10/(x + 3)
b) 2x⁴ - 4x³ + 5x² - 15x + 42 + 10/(x + 3)
c) 2x⁴ - 4x³ + 5x² + 14x + 41 + 10/(x + 3)
d) 2x⁴ - 4x³ + 5x² - 14x - 42 + 5/(x + 3)
9) (x⁵ + x³ + 3)/(x - 2)
a) x⁴ + 2x³ + 4x² + 9x + 18 + 39/(x - 2)
b) x⁴ + 3x² + 9/(x - 2)
c) x⁴ + 2x³ + 5x² + 10x + 20 + 43/(x - 2)
d) x⁴ + 3 + 9/(x - 2)
use the leading coefficient test to determine the end - behavior of the polynomial function.
10) f(x)=x³(x - 2)(x + 5)²
a) falls to the left and rises to the right
b) rises to the left and falls to the right
c) falls to the left and falls to the right
d) rises to the left and rises to the right
11) f(x)= - 6x³(x - 2)(x + 3)²
a) rises to the left and falls to the right
b) rises to the left and rises to the right
c) falls to the left and rises to the right
d) falls to the left and falls to the right

Question

Accepted Answer

### 1. Solve the equation $x^{2}-8x - 13=0$ by completing the square
# Explanation:
## Step1: Move the constant term
Add 13 to both sides of the equation:
$x^{2}-8x=13$
## Step2: Complete the square on the left - hand side
Take half of the coefficient of $x$, square it and add it to both sides. The coefficient of $x$ is - 8. Half of it is - 4, and its square is 16.
$x^{2}-8x + 16=13 + 16$
## Step3: Rewrite the left - hand side as a perfect square
$(x - 4)^{2}=29$
## Step4: Solve for $x$
Take the square root of both sides:
$x-4=\pm\sqrt{29}$
$x = 4\pm\sqrt{29}$
The solutions are $x=4+\sqrt{29}$ and $x=4 - \sqrt{29}$
# Answer:
D. $\{4-\sqrt{29},4+\sqrt{29}\}$

### 2. An object is propelled vertically upward from the top of a 144 - foot building
# Explanation:
## Step1: Set up the equation
The height function is $s(t)=-16t^{2}+176t + 144$. When the object hits the ground, $s(t)=0$. So we have the quadratic equation $-16t^{2}+176t + 144 = 0$. Divide through by - 16 to simplify: $t^{2}-11t - 9=0$
## Step2: Use the quadratic formula
The quadratic formula for $ax^{2}+bx + c = 0$ is $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 1$, $b=-11$, and $c=-9$
$t=\frac{11\pm\sqrt{(-11)^{2}-4	imes1	imes(-9)}}{2	imes1}=\frac{11\pm\sqrt{121 + 36}}{2}=\frac{11\pm\sqrt{157}}{2}$
$t=\frac{11\pm12.53}{2}$
We get two solutions: $t_1=\frac{11 + 12.53}{2}=11.765\approx11.8$ and $t_2=\frac{11-12.53}{2}<0$. We discard the negative solution.
# Answer:
C. 11.8 seconds

### 3. Write an expression for the area of the shaded region
# Explanation:
## Step1: Find the area of the large square and the small square
The area of the large square with side length $y$ is $A_{1}=y^{2}$. The area of the small square with side length 10 is $A_{2}=100$
## Step2: Find the area of the shaded region
The area of the shaded region $A=A_{1}-A_{2}=y^{2}-100$
## Step3: Factor the expression
Using the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$, where $a = y$ and $b = 10$, we get $A=(y + 10)(y - 10)$
# Answer:
A. $(y + 10)(y - 10)$

### 4. Claire has received scores of 85, 88, 87, and 80 on her algebra tests
# Explanation:
## Step1: Set up the average formula
Let the fifth - test score be $x$. The average of the five scores is $\frac{85 + 88+87 + 80+x}{5}$. We want this average to be at least 82, so $\frac{85 + 88+87 + 80+x}{5}\geq82$
## Step2: Simplify the left - hand side of the inequality
First, add the known scores: $85+88+87 + 80=340$. The inequality becomes $\frac{340+x}{5}\geq82$
## Step3: Solve the inequality
Multiply both sides by 5: $340+x\geq410$. Subtract 340 from both sides: $x\geq70$
# Answer:
A. 70 or greater

### 5. You have 200 feet of fencing to enclose a rectangular region
# Explanation:
## Step1: Let the length of the rectangle be $l$ and the width be $w$
The perimeter of a rectangle is $P = 2l+2w$. Given $P = 200$, so $2l+2w=200$, which simplifies to $l + w=100$, or $l=100 - w$
## Step2: Find the area function
The area of a rectangle is $A=lw=(100 - w)w=-w^{2}+100w$
## Step3: Find the vertex of the quadratic function
For a quadratic function $y = ax^{2}+bx + c$, the $x$ - coordinate of the vertex is $x=-\frac{b}{2a}$. Here, $a=-1$, $b = 100$. So $w=-\frac{100}{2	imes(-1)} = 50$
Since $l=100 - w$, then $l = 50$
# Answer:
The dimensions are 50 feet by 50 feet

### 6. A square sheet of paper measures 30 centimeters on each side
# Explanation:
## Step1: Use the Pythagorean theorem
For a square with side length $a$, the diagonal $d$ satisfies $d^{2}=a^{2}+a^{2}$ (since in a square, if the side length is $a$, and the diagonal forms the hypotenuse of a right - triangle with two sides of the square). Given $a = 30$, then $d^{2}=30^{2}+30^{2}=900+900 = 1800$
$d=\sqrt{1800}=30\sqrt{2}$
# Answer:
B. $30\sqrt{2}$ cm

### 7. Divide $5x^{2}-43x + 56$ by $x - 7$ using synthetic division
# Explanation:
## Step1: Set up synthetic division
The divisor is $x - 7$, so we use 7 in the synthetic - division setup. The coefficients of the dividend are 5, - 43, 56
## Step2: Perform synthetic division
Bring down the first coefficient 5:
Multiply 7 by 5 to get 35, add to - 43 to get - 8. Multiply 7 by - 8 to get - 56, add to 56 to get 0
The quotient is $5x-8$
# Answer:
A. $5x - 8$

### 8. Divide $(2x^{5}+2x^{4}-7x^{3}+x^{2}-x + 128)$ by $(x + 3)$
# Explanation:
## Step1: Use synthetic division
The divisor is $x + 3$, so we use - 3 in the synthetic - division setup. The coefficients of the dividend are 2, 2, - 7, 1, - 1, 128
## Step2: Perform synthetic division
Bring down the first coefficient 2:
Multiply - 3 by 2 to get - 6, add to 2 to get - 4. Multiply - 3 by - 4 to get 12, add to - 7 to get 5. Multiply - 3 by 5 to get - 15, add to 1 to get - 14. Multiply - 3 by - 14 to get 42, add to - 1 to get 41. Multiply - 3 by 41 to get - 123, add to 128 to get 5
The quotient is $2x^{4}-4x^{3}+5x^{2}-14x + 41$ with a remainder of 5, so the result is $2x^{4}-4x^{3}+5x^{2}-14x + 41+\frac{5}{x + 3}$
# Answer:
D. $2x^{4}-4x^{3}+5x^{2}-14x + 41+\frac{5}{x + 3}$

### 9. Divide $\frac{x^{5}+x^{3}+3}{x - 2}$
# Explanation:
## Step1: Use synthetic division
The divisor is $x - 2$, so we use 2 in the synthetic - division setup. The coefficients of the dividend are 1, 0, 1, 0, 0, 3
## Step2: Perform synthetic division
Bring down the first coefficient 1:
Multiply 2 by 1 to get 2, add to 0 to get 2. Multiply 2 by 2 to get 4, add to 1 to get 5. Multiply 2 by 5 to get 10, add to 0 to get 10. Multiply 2 by 10 to get 20, add to 0 to get 20. Multiply 2 by 20 to get 40, add to 3 to get 43
The quotient is $x^{4}+2x^{3}+5x^{2}+10x + 20$ with a remainder of 43, so the result is $x^{4}+2x^{3}+5x^{2}+10x + 20+\frac{43}{x - 2}$
# Answer:
C. $x^{4}+2x^{3}+5x^{2}+10x + 20+\frac{43}{x - 2}$

### 10. Use the Leading Coefficient Test for $f(x)=x^{3}(x - 2)(x + 5)^{2}$
# Explanation:
## Step1: Expand the polynomial (or analyze the degree and leading coefficient)
The degree of the polynomial is $3+1 + 2=6$ (a non - negative even degree) and the leading coefficient is positive (since the product of the leading coefficients of each factor is positive). For a polynomial $y = a_nx^n+\cdots+a_0$ with $n$ even and $a_n>0$, the graph rises to the left and rises to the right
# Answer:
B. rises to the left and rises to the right

### 11. Use the Leading Coefficient Test for $f(x)=-6x^{3}(x - 2)(x + 3)^{2}$
# Explanation:
## Step1: Analyze the degree and leading coefficient
The degree of the polynomial is $3+1+2 = 6$ (an even degree) and the leading coefficient is negative (because of the - 6). For a polynomial $y=a_nx^n+\cdots+a_0$ with $n$ even and $a_n<0$, the graph falls to the left and falls to the right
# Answer:
D. falls to the left and falls to the right

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QUESTION IMAGE

Question

Explanation:

1. Solve the equation $x^{2}-8x - 13=0$ by completing the square

Step1: Move the constant term

Step2: Complete the square on the left - hand side

Step3: Rewrite the left - hand side as a perfect square

Step4: Solve for $x$

Step1: Set up the equation

Step2: Use the quadratic formula

Step1: Find the area of the large square and the small square

Step2: Find the area of the shaded region

Step3: Factor the expression

Answer:

2. An object is propelled vertically upward from the top of a 144 - foot building