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
Which graph shows 3y - 2x = 6?
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A.
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B.
Correct Answer & Rationale
Correct Answer: B
To determine the graph of the equation \(3y - 2x = 6\), we can rearrange it into slope-intercept form \(y = mx + b\). This gives us \(y = \frac{2}{3}x + 2\), indicating a slope of \(\frac{2}{3}\) and a y-intercept of 2. Option B accurately represents this line, showing the correct slope and intercept. In contrast, Option A does not align with the expected slope or y-intercept, thus failing to represent the equation correctly. The visual representation in Option B confirms the relationship defined by the equation.
To determine the graph of the equation \(3y - 2x = 6\), we can rearrange it into slope-intercept form \(y = mx + b\). This gives us \(y = \frac{2}{3}x + 2\), indicating a slope of \(\frac{2}{3}\) and a y-intercept of 2. Option B accurately represents this line, showing the correct slope and intercept. In contrast, Option A does not align with the expected slope or y-intercept, thus failing to represent the equation correctly. The visual representation in Option B confirms the relationship defined by the equation.
A diver jumps from a platform. The height, h meters, the diver is above the water t seconds after jumping is represented by h = -16t^2 + 16t + 6.5. To the near hundredth of a second, how many seconds after jumping is the diver 2.5 meters above the water?
- A. 2.79
- B. 1.32
- C. 2.83
- D. 1.21
Correct Answer & Rationale
Correct Answer: D
To find when the diver is 2.5 meters above the water, substitute h = 2.5 into the equation: \[ 2.5 = -16t^2 + 16t + 6.5. \] Rearranging gives: \[ -16t^2 + 16t + 4 = 0. \] Using the quadratic formula, we solve for t, yielding two potential solutions. The option D (1.21 seconds) is valid as it falls within the realistic time frame of the jump. Options A (2.79) and C (2.83) exceed the expected time of descent, while B (1.32) does not satisfy the equation, confirming that only D accurately represents the diver's position at 2.5 meters above the water.
To find when the diver is 2.5 meters above the water, substitute h = 2.5 into the equation: \[ 2.5 = -16t^2 + 16t + 6.5. \] Rearranging gives: \[ -16t^2 + 16t + 4 = 0. \] Using the quadratic formula, we solve for t, yielding two potential solutions. The option D (1.21 seconds) is valid as it falls within the realistic time frame of the jump. Options A (2.79) and C (2.83) exceed the expected time of descent, while B (1.32) does not satisfy the equation, confirming that only D accurately represents the diver's position at 2.5 meters above the water.
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.
What is the value of 2/5 multiplied by 5/4 divided by 4/3
- A. 32/75
- B. 3\8
- C. 6\25
- D. 2\3
Correct Answer & Rationale
Correct Answer: B
To solve \( \frac{2}{5} \times \frac{5}{4} \div \frac{4}{3} \), we first multiply \( \frac{2}{5} \) by \( \frac{5}{4} \). This results in \( \frac{2 \times 5}{5 \times 4} = \frac{10}{20} = \frac{1}{2} \). Next, dividing by \( \frac{4}{3} \) is the same as multiplying by its reciprocal, \( \frac{3}{4} \). Therefore, \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \). Option A, \( \frac{32}{75} \), is incorrect as it does not simplify from the given operations. Option C, \( \frac{6}{25} \), results from miscalculating the division. Option D, \( \frac{2}{3} \), is also incorrect as it doesn't follow from the correct operations.
To solve \( \frac{2}{5} \times \frac{5}{4} \div \frac{4}{3} \), we first multiply \( \frac{2}{5} \) by \( \frac{5}{4} \). This results in \( \frac{2 \times 5}{5 \times 4} = \frac{10}{20} = \frac{1}{2} \). Next, dividing by \( \frac{4}{3} \) is the same as multiplying by its reciprocal, \( \frac{3}{4} \). Therefore, \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \). Option A, \( \frac{32}{75} \), is incorrect as it does not simplify from the given operations. Option C, \( \frac{6}{25} \), results from miscalculating the division. Option D, \( \frac{2}{3} \), is also incorrect as it doesn't follow from the correct operations.