module test
1. what is the factored form of the expression ( x^2 - 15x + 54 )?
a ( (x - 3)(x - 18) ) c ( (x - 6)(x - 9) )
b ( (x + 3)(x - 18) ) d ( (x + 6)(x - 9) )

2. what is the factored form of the expression ( 3x^2 - 10x - 8 )?
a ( (3x - 2)(x + 4) ) c ( (3x - 4)(x + 2) )
b ( (3x + 2)(x - 4) ) d ( (3x + 4)(x - 2) )

3. the graph of the function ( f(x) = -x^2 - 5x + 24 ) is shown below.
graph of a parabola with x-intercepts at approximately -8 and 3, y-intercept at 24
what are the approximate zeros of the function?
a ( -8 ) and ( 24 ) c ( 3 ) and ( 24 )
b ( -8 ) and ( -3 ) d ( 3 ) and ( -8 )

4. what are the points of intersection of the graphs of ( f(x) = 4x^2 - 5x + 6 ) and ( g(x) = 2x^2 - 3x + 18 )?
a ( (-2, -20) ) and ( (3, -45) )
b ( (-2, 32) ) and ( (3, 27) )
c ( (-3, -27) ) and ( (2, -32) )
d ( (-3, 45) ) and ( (2, 20) )

5. solve the equation ( 2x^2 - 24x + 72 = 0 ).
( x = )

6. what is the greatest common factor of the expression ( 28x^2y^4 - 35x^4y )?

7. solve the equation ( x^2 - 9x = 0 ).
( x = ) or ( x = )

8. what is a factored form of the expression ( 4x^4 - 48x^3 )?
a ( x^3(4x - 48) ) c ( 4x^3(x - 12) )
b ( x^4(4 - 48x) ) d ( 48x^3(4x - 1) )

9. what are the zeros of the function ( f(x) = 3x^2 - 75 )?
select all the correct answers.
a ( -5 ) d ( 3 )
b ( -3 ) e ( 5 )
c ( 0 )

10. what are the x-intercepts of the graph of the function ( f(x) = x^2 + 9x - 52 )?
and

11. what is the factored form of the expression ( 6x^2 - 5x - 21 )?

12. the area of a rectangle is ( 35x^2 - 75x ) square inches. in terms of ( x ), what is the length and width of the rectangle in inches?
length:
width:

Question

module test
1. what is the factored form of the expression ( x^2 - 15x + 54 )?
a ( (x - 3)(x - 18) )  c ( (x - 6)(x - 9) )
b ( (x + 3)(x - 18) )  d ( (x + 6)(x - 9) )

2. what is the factored form of the expression ( 3x^2 - 10x - 8 )?
a ( (3x - 2)(x + 4) )  c ( (3x - 4)(x + 2) )
b ( (3x + 2)(x - 4) )  d ( (3x + 4)(x - 2) )

3. the graph of the function ( f(x) = -x^2 - 5x + 24 ) is shown below.
graph of a parabola with x-intercepts at approximately -8 and 3, y-intercept at 24
what are the approximate zeros of the function?
a ( -8 ) and ( 24 )  c ( 3 ) and ( 24 )
b ( -8 ) and ( -3 )  d ( 3 ) and ( -8 )

4. what are the points of intersection of the graphs of ( f(x) = 4x^2 - 5x + 6 ) and ( g(x) = 2x^2 - 3x + 18 )?
a ( (-2, -20) ) and ( (3, -45) )
b ( (-2, 32) ) and ( (3, 27) )
c ( (-3, -27) ) and ( (2, -32) )
d ( (-3, 45) ) and ( (2, 20) )

5. solve the equation ( 2x^2 - 24x + 72 = 0 ).
( x = )

6. what is the greatest common factor of the expression ( 28x^2y^4 - 35x^4y )?

7. solve the equation ( x^2 - 9x = 0 ).
( x = ) or ( x = )

8. what is a factored form of the expression ( 4x^4 - 48x^3 )?
a ( x^3(4x - 48) )  c ( 4x^3(x - 12) )
b ( x^4(4 - 48x) )  d ( 48x^3(4x - 1) )

9. what are the zeros of the function ( f(x) = 3x^2 - 75 )?
select all the correct answers.
a ( -5 )  d ( 3 )
b ( -3 )  e ( 5 )
c ( 0 )

10. what are the x-intercepts of the graph of the function ( f(x) = x^2 + 9x - 52 )?
 and

11. what is the factored form of the expression ( 6x^2 - 5x - 21 )?

12. the area of a rectangle is ( 35x^2 - 75x ) square inches. in terms of ( x ), what is the length and width of the rectangle in inches?
length:
width:

Accepted Answer

### Question 1
# Explanation:
## Step1: Find two numbers that multiply to 54 and add to -15.
The numbers are -6 and -9 since $(-6)\times(-9) = 54$ and $(-6)+(-9)=-15$.
## Step2: Write the factored form.
So $x^{2}-15x + 54=(x - 6)(x - 9)$.
# Answer: C. $(x - 6)(x - 9)$

### Question 2
# Explanation:
## Step1: Use the AC method. For $3x^{2}-10x - 8$, $A = 3$, $B=-10$, $C=-8$. $AC=3\times(-8)=-24$. Find two numbers that multiply to -24 and add to -10. The numbers are -12 and 2.
## Step2: Rewrite the middle term: $3x^{2}-12x+2x - 8$.
## Step3: Factor by grouping: $3x(x - 4)+2(x - 4)=(3x + 2)(x - 4)$.
# Answer: B. $(3x + 2)(x - 4)$

### Question 3
# Explanation:
The zeros of the function are the x - intercepts. From the graph, we can see that the graph crosses the x - axis at $x=-8$ and $x = 3$ (by looking at the grid and the intersection points).
# Answer: D. 3 and -8

### Question 4
# Explanation:
## Step1: Set $f(x)=g(x)$: $4x^{2}-5x + 6=2x^{2}-3x + 18$.
## Step2: Subtract $2x^{2}-3x + 18$ from both sides: $2x^{2}-2x - 12 = 0$. Divide by 2: $x^{2}-x - 6=0$.
## Step3: Factor: $(x - 3)(x + 2)=0$. So $x = 3$ or $x=-2$.
## Step4: Find the y - values. For $x=-2$, $f(-2)=4\times(-2)^{2}-5\times(-2)+6=16 + 10+6 = 32$. For $x = 3$, $f(3)=4\times3^{2}-5\times3+6=36-15 + 6=27$. So the points are $(-2,32)$ and $(3,27)$.
# Answer: B. $(-2, 32)$ and $(3, 27)$

### Question 5
# Explanation:
## Step1: Divide the equation $2x^{2}-24x + 72 = 0$ by 2: $x^{2}-12x + 36=0$.
## Step2: Factor: $(x - 6)^{2}=0$. So $x = 6$.
# Answer: $x = 6$

### Question 6
# Explanation:
## Step1: Find the GCF of the coefficients and the variables. For $28x^{2}y^{4}-35x^{4}y$, the GCF of 28 and 35 is 7. The GCF of $x^{2}$ and $x^{4}$ is $x^{2}$, and the GCF of $y^{4}$ and $y$ is $y$.
## Step2: Factor out the GCF: $7x^{2}y(4y^{3}-5x^{2})$. Wait, maybe I made a mistake. Wait, the original expression is $28x^{2}y^{4}-35x^{4}y$. Let's re - calculate. The GCF of 28 and 35 is 7, GCF of $x^{2}$ and $x^{4}$ is $x^{2}$, GCF of $y^{4}$ and $y$ is $y$. So factoring out $7x^{2}y$ gives $7x^{2}y(4y^{3}-5x^{2})$. But maybe the question is to factor the numerical and variable parts separately. Wait, the GCF of 28 and 35 is 7, GCF of $x^{2}$ and $x^{4}$ is $x^{2}$, GCF of $y^{4}$ and $y$ is $y$. So the GCF is $7x^{2}y$.
# Answer: The greatest common factor is $7x^{2}y$

### Question 7
# Explanation:
## Step1: Factor the equation $x^{2}-9x = 0$ as $x(x - 9)=0$.
## Step2: Set each factor equal to zero: $x = 0$ or $x-9=0$. So $x = 0$ or $x = 9$.
# Answer: $x = 0$ or $x = 9$

### Question 8
# Explanation:
## Step1: Factor out the GCF of $4x^{4}-48x^{3}$. The GCF of 4 and 48 is 4, and the GCF of $x^{4}$ and $x^{3}$ is $x^{3}$.
## Step2: Factor out $4x^{3}$: $4x^{3}(x - 12)$.
# Answer: C. $4x^{3}(x - 12)$

### Question 9
# Explanation:
## Step1: Factor the function $f(x)=3x^{2}-75$. Factor out 3: $f(x)=3(x^{2}-25)$.
## Step2: Use the difference of squares: $x^{2}-25=(x - 5)(x + 5)$. So $f(x)=3(x - 5)(x + 5)$.
## Step3: Set $f(x)=0$: $3(x - 5)(x + 5)=0$. So $x = 5$ or $x=-5$.
# Answer: A. -5, E. 5

### Question 10
# Explanation:
## Step1: Factor the equation $x^{2}+9x - 52 = 0$. Find two numbers that multiply to -52 and add to 9. The numbers are 13 and -4.
## Step2: So $x^{2}+9x - 52=(x + 13)(x - 4)=0$.
## Step3: Set each factor equal to zero: $x=-13$ or $x = 4$.
# Answer: -13 and 4

### Question 11
# Explanation:
## Step1: Use the AC method for $6x^{2}-5x - 21$. $A = 6$, $B=-5$, $C=-21$. $AC=6\times(-21)=-126$. Find two numbers that multiply to -126 and add to -5. The numbers are -14 and 9.
## Step2: Rewrite the middle term: $6x^{2}+9x-14x - 21$.
## Step3: Factor by grouping: $3x(2x + 3)-7(2x + 3)=(3x - 7)(2x + 3)$. Wait, let's check: $(3x - 7)(2x+3)=6x^{2}+9x-14x - 21=6x^{2}-5x - 21$. Yes.
# Answer: $(3x - 7)(2x + 3)$ (or $(2x + 3)(3x - 7)$)

### Question 12
# Explanation:
## Step1: Factor the area expression $35x^{2}-75x$. Factor out 5x: $5x(7x - 15)$.
## Step2: So the length and width can be $5x$ and $7x - 15$ (or vice - versa, since length and width are interchangeable for a rectangle).
# Answer:
Length: $5x$ inches (or $7x - 15$ inches)
Width: $7x - 15$ inches (or $5x$ inches)

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QUESTION IMAGE

Question

Explanation:

Question 1

Step1: Find two numbers that multiply to 54 and add to -15.

Step2: Write the factored form.

Step1: Use the AC method. For \(3x^{2}-10x - 8\), \(A = 3\), \(B=-10\), \(C=-8\). \(AC=3\times(-8)=-24\). Find two numbers that multiply to -24 and add to -10. The numbers are -12 and 2.

Step2: Rewrite the middle term: \(3x^{2}-12x+2x - 8\).

Step3: Factor by grouping: \(3x(x - 4)+2(x - 4)=(3x + 2)(x - 4)\).

Answer:

Question 2