Factor the expression completely: 45bcx - 10ax
- A. 5x(9bc - 2a)
- B. 5(9bc - 2a)
- C. x(45bc - 10a)
- D. 5x(9bc + 2a)
Correct Answer & Rationale
Correct Answer: A
To factor the expression 45bcx - 10ax completely, we start by identifying the greatest common factor (GCF). The GCF of the coefficients 45 and 10 is 5, and both terms contain the variable x. Thus, we can factor out 5x, resulting in 5x(9bc - 2a). Option A accurately reflects this factorization. Option B lacks the variable x, which is essential in the original expression. Option C incorrectly factors out only x, missing the GCF of 5. Option D alters the sign of the second term, which does not represent the original expression correctly.
To factor the expression 45bcx - 10ax completely, we start by identifying the greatest common factor (GCF). The GCF of the coefficients 45 and 10 is 5, and both terms contain the variable x. Thus, we can factor out 5x, resulting in 5x(9bc - 2a). Option A accurately reflects this factorization. Option B lacks the variable x, which is essential in the original expression. Option C incorrectly factors out only x, missing the GCF of 5. Option D alters the sign of the second term, which does not represent the original expression correctly.
Other Related Questions
On a number line, what is the distance, in units, between 16 and -25
Correct Answer & Rationale
Correct Answer: 41 units
To find the distance between two points on a number line, subtract the smaller number from the larger number. Here, the calculation is |16 - (-25)|, which simplifies to |16 + 25| = |41|. This results in a distance of 41 units. Other options may suggest incorrect calculations. For instance, an answer like 9 units might arise from simply adding the two numbers without considering their positions on the number line, leading to an inaccurate interpretation of distance. Similarly, options like 25 or 16 units misrepresent the actual distance by not accounting for both numbers' magnitudes relative to zero.
To find the distance between two points on a number line, subtract the smaller number from the larger number. Here, the calculation is |16 - (-25)|, which simplifies to |16 + 25| = |41|. This results in a distance of 41 units. Other options may suggest incorrect calculations. For instance, an answer like 9 units might arise from simply adding the two numbers without considering their positions on the number line, leading to an inaccurate interpretation of distance. Similarly, options like 25 or 16 units misrepresent the actual distance by not accounting for both numbers' magnitudes relative to zero.
Which graph represents a function?
- A. M-27A.png
- B. M-27B.png
- C. M-27C.png
Correct Answer & Rationale
Correct Answer: B
To determine which graph represents a function, we apply the Vertical Line Test. This test states that if a vertical line intersects the graph at more than one point, the graph does not represent a function. Option A fails this test, as a vertical line can intersect the graph at multiple points, indicating it is not a function. Option C also does not satisfy the criteria, showing multiple intersections with vertical lines. In contrast, Option B passes the Vertical Line Test, as any vertical line drawn will intersect the graph at only one point, confirming it represents a function.
To determine which graph represents a function, we apply the Vertical Line Test. This test states that if a vertical line intersects the graph at more than one point, the graph does not represent a function. Option A fails this test, as a vertical line can intersect the graph at multiple points, indicating it is not a function. Option C also does not satisfy the criteria, showing multiple intersections with vertical lines. In contrast, Option B passes the Vertical Line Test, as any vertical line drawn will intersect the graph at only one point, confirming it represents a function.
An advertisement poster in the window of a shoe store is in the shape of a rectangle. The length of the poster is 9 less than 4 times the width. Which expression represents the length of the poster when w is the width
- A. 4w - 9
- B. 9 - 4w
- C. 4w + 9
- D. 9w - 4
Correct Answer & Rationale
Correct Answer: A
The expression for the length of the poster is determined by the relationship given in the problem. The length is described as "9 less than 4 times the width," which translates mathematically to \(4w - 9\). Option A (4w - 9) accurately reflects this relationship. Option B (9 - 4w) incorrectly suggests that the length is greater than 9 and decreases as width increases, which contradicts the problem's description. Option C (4w + 9) implies that the length increases by 9, rather than decreasing, which is not aligned with the original statement. Option D (9w - 4) introduces an incorrect multiplication factor and does not adhere to the given relationship, making it invalid.
The expression for the length of the poster is determined by the relationship given in the problem. The length is described as "9 less than 4 times the width," which translates mathematically to \(4w - 9\). Option A (4w - 9) accurately reflects this relationship. Option B (9 - 4w) incorrectly suggests that the length is greater than 9 and decreases as width increases, which contradicts the problem's description. Option C (4w + 9) implies that the length increases by 9, rather than decreasing, which is not aligned with the original statement. Option D (9w - 4) introduces an incorrect multiplication factor and does not adhere to the given relationship, making it invalid.
The distance, d, in feet, it takes to come to a complete stop when driving a car r miles per hour can be found using the equation d = 1/20(r^2)+ r. If it takes a car 240 feet to come to a complete stop, what was the speed of the car, in miles per hour, when the driver began to stop it?
- A. 40
- B. 30
- C. 60
- D. 80
Correct Answer & Rationale
Correct Answer: A
To find the speed of the car when it takes 240 feet to stop, substitute d = 240 into the equation d = 1/20(r^2) + r. This leads to the equation 240 = 1/20(r^2) + r. Multiplying through by 20 simplifies to 4800 = r^2 + 20r, which rearranges to r^2 + 20r - 4800 = 0. Solving this quadratic equation yields r = 40 or r = -120. Since speed cannot be negative, the valid solution is 40 mph. Option B (30) does not satisfy the equation, leading to a shorter stopping distance. Option C (60) results in a stopping distance of 480 feet, which exceeds 240 feet. Option D (80) produces a stopping distance of 800 feet, also incorrect. Thus, only 40 mph meets the criteria.
To find the speed of the car when it takes 240 feet to stop, substitute d = 240 into the equation d = 1/20(r^2) + r. This leads to the equation 240 = 1/20(r^2) + r. Multiplying through by 20 simplifies to 4800 = r^2 + 20r, which rearranges to r^2 + 20r - 4800 = 0. Solving this quadratic equation yields r = 40 or r = -120. Since speed cannot be negative, the valid solution is 40 mph. Option B (30) does not satisfy the equation, leading to a shorter stopping distance. Option C (60) results in a stopping distance of 480 feet, which exceeds 240 feet. Option D (80) produces a stopping distance of 800 feet, also incorrect. Thus, only 40 mph meets the criteria.