The graph of the equation y = x^2 + 4x - 5 is shown on the grid. Which statement is true when y = 0?
- A. x= -5 and x=1
- B. x= -2
- C. x= -5 and x = 0
- D. x= -9
Correct Answer & Rationale
Correct Answer: A
To find the values of x when y = 0, we need to solve the equation \(x^2 + 4x - 5 = 0\). Factoring this quadratic gives \((x + 5)(x - 1) = 0\), leading to the solutions \(x = -5\) and \(x = 1\). Option A correctly identifies these solutions. Option B states \(x = -2\), which is not a solution to the equation. Option C suggests \(x = -5\) and \(x = 0\); while it includes one correct solution, \(x = 0\) is incorrect. Option D claims \(x = -9\), which does not satisfy the equation. Thus, only option A accurately reflects the solutions when y = 0.
To find the values of x when y = 0, we need to solve the equation \(x^2 + 4x - 5 = 0\). Factoring this quadratic gives \((x + 5)(x - 1) = 0\), leading to the solutions \(x = -5\) and \(x = 1\). Option A correctly identifies these solutions. Option B states \(x = -2\), which is not a solution to the equation. Option C suggests \(x = -5\) and \(x = 0\); while it includes one correct solution, \(x = 0\) is incorrect. Option D claims \(x = -9\), which does not satisfy the equation. Thus, only option A accurately reflects the solutions when y = 0.
Other Related Questions
Last weekend, 625 runners entered a 10,000-meter race. A 10,000- meter race is 6.2 miles long. Ruben won the race with a finishing time of 29 minutes 51 seconds.
The graphs show information about the top 10 runners.
Based on the scatter plot, what is the range of ages of the top 10 runners?
- A. 9
- B. 1
- C. 16
- D. 40
Correct Answer & Rationale
Correct Answer: C
The range of ages is determined by subtracting the youngest runner's age from the oldest runner's age. In this case, the scatter plot indicates that the youngest runner is 16 years old and the oldest is 32 years old. Thus, the range is 32 - 16 = 16 years. Option A (9) incorrectly suggests a smaller age difference, while B (1) implies almost no age variation, neither of which aligns with the data presented. Option D (40) overestimates the age range, indicating a misunderstanding of the plotted values. Therefore, the accurate calculation of 16 years reflects the true age span of the top 10 runners.
The range of ages is determined by subtracting the youngest runner's age from the oldest runner's age. In this case, the scatter plot indicates that the youngest runner is 16 years old and the oldest is 32 years old. Thus, the range is 32 - 16 = 16 years. Option A (9) incorrectly suggests a smaller age difference, while B (1) implies almost no age variation, neither of which aligns with the data presented. Option D (40) overestimates the age range, indicating a misunderstanding of the plotted values. Therefore, the accurate calculation of 16 years reflects the true age span of the top 10 runners.
Which pair of equations represents parallel lines?
- A. -2x + y + 2 = 0, y = -(1/2)x - 4
- B. 3x + y = -8, y = 3x - 8
- C. x + 2y = 8, -x - 2y = 3
- D. -(2/3)x + y = 12, y = -(3/2)x - 1
Correct Answer & Rationale
Correct Answer: C
To identify parallel lines, the slopes of the equations must be equal. Option A has slopes of 1/2 and -1/2, which are not equal. Option B has slopes of 3 and 3, indicating the lines are parallel; however, it is not the correct answer as it does not match the requirement for both equations. Option C has the first equation rearranged to slope -1/2 and the second to slope -1/2, confirming they are parallel. Option D features slopes of 2/3 and -3/2, which are also not equal, indicating the lines intersect. Thus, only option C accurately represents parallel lines.
To identify parallel lines, the slopes of the equations must be equal. Option A has slopes of 1/2 and -1/2, which are not equal. Option B has slopes of 3 and 3, indicating the lines are parallel; however, it is not the correct answer as it does not match the requirement for both equations. Option C has the first equation rearranged to slope -1/2 and the second to slope -1/2, confirming they are parallel. Option D features slopes of 2/3 and -3/2, which are also not equal, indicating the lines intersect. Thus, only option C accurately represents parallel lines.
Select the factors for the following expression 2x^2 - xy - 3y^2
- A. (2x+3y)(x-y)
- B. (x+y)(2x-3y)
- C. (2x-y)(x+3y)
- D. (2x-3y)(x+y)
Correct Answer & Rationale
Correct Answer: D
To factor the expression \(2x^2 - xy - 3y^2\), we look for two binomials that multiply to give the original expression. Option D, \((2x-3y)(x+y)\), expands to \(2x^2 + 2xy - 3xy - 3y^2\), which simplifies to \(2x^2 - xy - 3y^2\), matching the original expression. Option A, \((2x+3y)(x-y)\), expands to \(2x^2 - 2xy + 3xy - 3y^2\), resulting in \(2x^2 + xy - 3y^2\), which is incorrect. Option B, \((x+y)(2x-3y)\), gives \(2x^2 - 3xy + 2xy - 3y^2\), simplifying to \(2x^2 - xy - 3y^2\), but the signs do not match the original expression. Option C, \((2x-y)(x+3y)\), expands to \(2x^2 + 6xy - xy - 3y^2\), leading to \(2x^2 + 5xy - 3y^2\), which is also incorrect. Thus, only Option D correctly factors the expression.
To factor the expression \(2x^2 - xy - 3y^2\), we look for two binomials that multiply to give the original expression. Option D, \((2x-3y)(x+y)\), expands to \(2x^2 + 2xy - 3xy - 3y^2\), which simplifies to \(2x^2 - xy - 3y^2\), matching the original expression. Option A, \((2x+3y)(x-y)\), expands to \(2x^2 - 2xy + 3xy - 3y^2\), resulting in \(2x^2 + xy - 3y^2\), which is incorrect. Option B, \((x+y)(2x-3y)\), gives \(2x^2 - 3xy + 2xy - 3y^2\), simplifying to \(2x^2 - xy - 3y^2\), but the signs do not match the original expression. Option C, \((2x-y)(x+3y)\), expands to \(2x^2 + 6xy - xy - 3y^2\), leading to \(2x^2 + 5xy - 3y^2\), which is also incorrect. Thus, only Option D correctly factors the expression.
Which list shows the numbers arranged from least to greatest?
- A. -(2/9), -0.21, -0.2, -(2/11), -1
- B. -1, -(2/9), -0.21, -0.2, -(2/11)
- C. -1, -(2/11), -0.21, -0.2, -(2/9)
- D. -(2/11), -0.2, -0.21, -(2/9), -1
Correct Answer & Rationale
Correct Answer: C
To determine the correct order, it's essential to convert fractions and decimals to comparable values. In option C, the numbers arranged from least to greatest are: -1, approximately -0.1818 (for -(2/11)), -0.21, -0.2, and approximately -0.2222 (for -(2/9)). This sequence accurately reflects their values. Option A incorrectly places -1 at the end, misordering the fractions and decimals. Option B also misplaces -1, and the order of the decimals is incorrect. Option D incorrectly ranks -1 as the least value and misplaces the fraction values, leading to an inaccurate arrangement.
To determine the correct order, it's essential to convert fractions and decimals to comparable values. In option C, the numbers arranged from least to greatest are: -1, approximately -0.1818 (for -(2/11)), -0.21, -0.2, and approximately -0.2222 (for -(2/9)). This sequence accurately reflects their values. Option A incorrectly places -1 at the end, misordering the fractions and decimals. Option B also misplaces -1, and the order of the decimals is incorrect. Option D incorrectly ranks -1 as the least value and misplaces the fraction values, leading to an inaccurate arrangement.