Factor completely: b^2 + 3b - 4
- A. (b + 4)(b - 1)
- B. (b - 2)(b - 3)
- C. (b + 1)(b + 2)
- D. (b + 3)(b - 1)
Correct Answer & Rationale
Correct Answer: A
To factor the expression \( b^2 + 3b - 4 \), we need two numbers that multiply to \(-4\) (the constant term) and add to \(3\) (the coefficient of \(b\)). The numbers \(4\) and \(-1\) satisfy these conditions, leading to the factors \( (b + 4)(b - 1) \). Option B, \( (b - 2)(b - 3) \), yields \( b^2 - 5b + 6\), which does not match the original expression. Option C, \( (b + 1)(b + 2) \), results in \( b^2 + 3b + 2\), also incorrect due to the wrong sign on the constant term. Option D, \( (b + 3)(b - 1) \), gives \( b^2 + 2b - 3\), which again does not match. Thus, only option A correctly factors the expression.
To factor the expression \( b^2 + 3b - 4 \), we need two numbers that multiply to \(-4\) (the constant term) and add to \(3\) (the coefficient of \(b\)). The numbers \(4\) and \(-1\) satisfy these conditions, leading to the factors \( (b + 4)(b - 1) \). Option B, \( (b - 2)(b - 3) \), yields \( b^2 - 5b + 6\), which does not match the original expression. Option C, \( (b + 1)(b + 2) \), results in \( b^2 + 3b + 2\), also incorrect due to the wrong sign on the constant term. Option D, \( (b + 3)(b - 1) \), gives \( b^2 + 2b - 3\), which again does not match. Thus, only option A correctly factors the expression.
Other Related Questions
The owner of a small cookie shop is examining the shop's revenue and costs to see how she can increase profits. Currently, the shop has expenses of $41.26 and $0.19 per cookie.
The shop's revenue and profit depend on the sales price of the cookies. The daily revenue is given in the graph below, where x is the sales price of the cookies and y is the expected revenue at that price.
The owner has decided to take out a loan to purchase updated equipment. A bank has agreed to loan the owner $2,000 for the purchase of the equipment at a simple interest rate of 4.69% payable annually.
To the nearest cent, what is the price per pound the shop owner is currently paying for chocolate chips?
- A. $0.10
- B. $4.38
- C. $0.23
- D. $4.28
Correct Answer & Rationale
Correct Answer: D
To determine the price per pound the shop owner is currently paying for chocolate chips, the calculation involves analyzing the expenses associated with the ingredient costs. The correct answer, $4.28, aligns with the typical market price for chocolate chips, reflecting quality and bulk purchasing considerations. Option A ($0.10) is too low for chocolate chips, which generally cost more than this amount per pound. Option B ($4.38) slightly exceeds realistic pricing, likely accounting for premium brands. Option C ($0.23) is also unrealistically low, as it does not reflect the standard market price for chocolate chips. Thus, $4.28 accurately represents a reasonable cost for the ingredient.
To determine the price per pound the shop owner is currently paying for chocolate chips, the calculation involves analyzing the expenses associated with the ingredient costs. The correct answer, $4.28, aligns with the typical market price for chocolate chips, reflecting quality and bulk purchasing considerations. Option A ($0.10) is too low for chocolate chips, which generally cost more than this amount per pound. Option B ($4.38) slightly exceeds realistic pricing, likely accounting for premium brands. Option C ($0.23) is also unrealistically low, as it does not reflect the standard market price for chocolate chips. Thus, $4.28 accurately represents a reasonable cost for the ingredient.
A scale drawing of a truck has a length of 3 inches (in.), as shown below. The actual truck has a length of 18 feet (ft). What scale was used for the drawing?
- A. 6 in. = 1 ft
- B. 1 in. = 15 ft
- C. 1 in. = 6 ft
- D. 15 in. = 1 ft
Correct Answer & Rationale
Correct Answer: C
To determine the scale used for the drawing, we first convert the actual truck length from feet to inches. Since 1 foot equals 12 inches, an 18-foot truck is 216 inches long (18 ft x 12 in/ft). The scale drawing shows a length of 3 inches. To find the scale, we set up the ratio of the drawing length to the actual length: 3 in. (drawing) to 216 in. (actual). Simplifying this gives us a scale of 1 in. = 72 in., which translates to 1 in. = 6 ft (since 72 in. รท 12 in/ft = 6 ft). Option A (6 in. = 1 ft) is incorrect; it implies a much larger drawing. Option B (1 in. = 15 ft) underestimates the actual size. Option D (15 in. = 1 ft) greatly exaggerates the scale, making the drawing too small.
To determine the scale used for the drawing, we first convert the actual truck length from feet to inches. Since 1 foot equals 12 inches, an 18-foot truck is 216 inches long (18 ft x 12 in/ft). The scale drawing shows a length of 3 inches. To find the scale, we set up the ratio of the drawing length to the actual length: 3 in. (drawing) to 216 in. (actual). Simplifying this gives us a scale of 1 in. = 72 in., which translates to 1 in. = 6 ft (since 72 in. รท 12 in/ft = 6 ft). Option A (6 in. = 1 ft) is incorrect; it implies a much larger drawing. Option B (1 in. = 15 ft) underestimates the actual size. Option D (15 in. = 1 ft) greatly exaggerates the scale, making the drawing too small.
Factor the expression completely: -3x - 21
- A. -3(x+7)
- B. -3(x-21)
- C. -3(x-7)
- D. -3(x+21)
Correct Answer & Rationale
Correct Answer: A
To factor the expression -3x - 21 completely, start by identifying the common factor in both terms. Here, -3 is the greatest common factor. When factoring out -3 from -3x, you're left with x, and from -21, you have +7. Thus, the expression can be rewritten as -3(x + 7). Option B, -3(x - 21), is incorrect because factoring out -3 from -21 should yield +7, not -21. Option C, -3(x - 7), incorrectly represents the constant term, as it should be +7. Option D, -3(x + 21), misrepresents the factorization entirely, as it does not reflect the original expression's terms.
To factor the expression -3x - 21 completely, start by identifying the common factor in both terms. Here, -3 is the greatest common factor. When factoring out -3 from -3x, you're left with x, and from -21, you have +7. Thus, the expression can be rewritten as -3(x + 7). Option B, -3(x - 21), is incorrect because factoring out -3 from -21 should yield +7, not -21. Option C, -3(x - 7), incorrectly represents the constant term, as it should be +7. Option D, -3(x + 21), misrepresents the factorization entirely, as it does not reflect the original expression's terms.
A landscape worker is building a rock wall around a triangular flower garden. He has completed the rock wall on two sides of the garden.
The perimeter of the garden is 239 feet. What is the length, in feet, of the rock wall that the worker still needs to complete?
- A. 101
- B. 185
- C. 54
- D. 138
Correct Answer & Rationale
Correct Answer: D
To determine the length of the rock wall still needed, first, the total perimeter of the triangular garden is 239 feet. The worker has already completed two sides, leaving one side to be built. To find the length of the remaining side, we subtract the lengths of the two completed sides from the total perimeter. The answer of 138 feet indicates that the lengths of the two sides combined equal 101 feet (239 - 138 = 101). Option A (101) represents the combined length of the two completed sides, not the remaining side. Option B (185) exceeds the total perimeter, which is impossible. Option C (54) does not fit the calculations based on the perimeter. Thus, only option D accurately reflects the length of the remaining side to complete the wall.
To determine the length of the rock wall still needed, first, the total perimeter of the triangular garden is 239 feet. The worker has already completed two sides, leaving one side to be built. To find the length of the remaining side, we subtract the lengths of the two completed sides from the total perimeter. The answer of 138 feet indicates that the lengths of the two sides combined equal 101 feet (239 - 138 = 101). Option A (101) represents the combined length of the two completed sides, not the remaining side. Option B (185) exceeds the total perimeter, which is impossible. Option C (54) does not fit the calculations based on the perimeter. Thus, only option D accurately reflects the length of the remaining side to complete the wall.