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
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.
The width of a painting is 24 centimeters shorter than its length, x. The area of the painting is 4,081 square centimeters. Which equation could be used to find the dimensions of the painting?
- A. x^2 - 24x - 4,081 = 0
- B. x^2 + 24x - 4,081 = 0
- C. x^2 + 24x + 4,081 = 0
- D. x^2 - 24x + 4,081 = 0
Correct Answer & Rationale
Correct Answer: A
To find the dimensions of the painting, we start with the relationship between length and width. The width is 24 cm shorter than the length \(x\), so it can be expressed as \(x - 24\). The area of a rectangle is given by the product of its length and width, resulting in the equation \(x(x - 24) = 4,081\). Expanding this leads to \(x^2 - 24x - 4,081 = 0\), which matches option A. Option B incorrectly adds 24x, leading to an incorrect area calculation. Option C incorrectly adds 24 and includes a positive constant, which does not represent the area. Option D incorrectly adds 4,081 and has a positive term that does not reflect the relationship between length and width.
To find the dimensions of the painting, we start with the relationship between length and width. The width is 24 cm shorter than the length \(x\), so it can be expressed as \(x - 24\). The area of a rectangle is given by the product of its length and width, resulting in the equation \(x(x - 24) = 4,081\). Expanding this leads to \(x^2 - 24x - 4,081 = 0\), which matches option A. Option B incorrectly adds 24x, leading to an incorrect area calculation. Option C incorrectly adds 24 and includes a positive constant, which does not represent the area. Option D incorrectly adds 4,081 and has a positive term that does not reflect the relationship between length and width.
An expression for a company's cost to make n bicycles is -0.017n? - 6.8n + 690. An expression for the revenue from selling these n bicycles is 70n. Profit is revenue minus cost. Which is an expression for the profit for making and selling n bicycles?
- A. -0.017n^2 - 76.8n + 690
- B. 0.017n^2 + 76.8n - 690
- C. 0.017n^2 + 63.2n + 690
- D. -0.017n^2 + 63.2n + 690
Correct Answer & Rationale
Correct Answer: D
To find the profit from selling n bicycles, subtract the cost expression from the revenue expression. The cost is given as -0.017n² - 6.8n + 690, and the revenue is 70n. Calculating profit: Profit = Revenue - Cost = 70n - (-0.017n² - 6.8n + 690) simplifies to 70n + 0.017n² + 6.8n - 690, which results in 0.017n² + 63.2n - 690. Option D, -0.017n² + 63.2n + 690, incorrectly presents the quadratic term with the wrong sign. Options A and B incorrectly combine terms or misrepresent the coefficients. Option C miscalculates the constant term. Thus, only option D maintains the correct profit structure.
To find the profit from selling n bicycles, subtract the cost expression from the revenue expression. The cost is given as -0.017n² - 6.8n + 690, and the revenue is 70n. Calculating profit: Profit = Revenue - Cost = 70n - (-0.017n² - 6.8n + 690) simplifies to 70n + 0.017n² + 6.8n - 690, which results in 0.017n² + 63.2n - 690. Option D, -0.017n² + 63.2n + 690, incorrectly presents the quadratic term with the wrong sign. Options A and B incorrectly combine terms or misrepresent the coefficients. Option C miscalculates the constant term. Thus, only option D maintains the correct profit structure.
Which expression is equivalent to (3a + 4ab - 7b) - (a + 2ab - 4b)?
- A. 2a + 2ab - 11b
- B. 2a + 6ab - 11b
- C. 2a + 2ab - 3b
- D. 2a + 6ab - 35
Correct Answer & Rationale
Correct Answer: C
To simplify the expression \((3a + 4ab - 7b) - (a + 2ab - 4b)\), start by distributing the negative sign across the second set of parentheses: \[ 3a + 4ab - 7b - a - 2ab + 4b \] Next, combine like terms: - For \(a\): \(3a - a = 2a\) - For \(ab\): \(4ab - 2ab = 2ab\) - For \(b\): \(-7b + 4b = -3b\) This results in the expression \(2a + 2ab - 3b\), matching option C. Option A introduces an incorrect coefficient for \(b\), while option B miscalculates the \(ab\) term. Option D incorrectly combines terms, leading to an erroneous constant. Thus, option C is the only accurate simplification.
To simplify the expression \((3a + 4ab - 7b) - (a + 2ab - 4b)\), start by distributing the negative sign across the second set of parentheses: \[ 3a + 4ab - 7b - a - 2ab + 4b \] Next, combine like terms: - For \(a\): \(3a - a = 2a\) - For \(ab\): \(4ab - 2ab = 2ab\) - For \(b\): \(-7b + 4b = -3b\) This results in the expression \(2a + 2ab - 3b\), matching option C. Option A introduces an incorrect coefficient for \(b\), while option B miscalculates the \(ab\) term. Option D incorrectly combines terms, leading to an erroneous constant. Thus, option C is the only accurate simplification.