12345 - Math Quiz -Algebra Covering: linear equations, inequalities, systems of equations, absolute value expressions, and quadratic equations. 1 / 5 1. Solve the following equation for x: 3xÂ² - 7x + 2 = 0. a) x = 1, x = 2 b) x = -1, x = -2 c) x = 1/3, x = 2 d) x = -1/3, x = 2 To factor the given quadratic equation, we look for two numbers that multiply to 6 (3 multiplied by the constant term 2) and add up to -7 (the coefficient of the middle term). The two numbers that fit these criteria are -6 and -1. Now, we can rewrite the equation by factoring the quadratic expression: 3xÂ² - 7x + 2 = (3x - 1)(x - 2) = 0 We then set each factor equal to zero to solve for x: 3x - 1 = 0 => x = 1/3 x - 2 = 0 => x = 2 Thus, the solution to the quadratic equation 3xÂ² - 7x + 2 = 0 is x = 1/3 and x = 2, which corresponds to option C. 2 / 5 2. The sum of two numbers is 24, and their difference is 10. What are the two numbers? a) 10, 14 b) 12, 12 c) 17, 7 d) 13, 11 Let x and y be the two numbers. Write the system of linear equations using the given information: x + y = 24 x - y = 10 Add the two equations together to eliminate y: 2x = 34 Divide by 2 to solve for x: x = 17 Now substitute x into the first equation: 17 + y = 24 Subtract 17 from both sides of the equation to solve for y: y = 7 Thus, the two numbers are 17 and 7, which corresponds to option C. 3 / 5 3. What is the inverse of the function f(x) = 3x - 2? a) f^(-1)(x) = 3x + 2 b) f^(-1)(x) = 3x - 2 c) f^(-1)(x) = (x + 2) / 3 d) f^(-1)(x) = (x - 2) / 3 To find the inverse of a function, we swap the x and y coordinates (x and f(x), respectively) and then solve for the new f(x): 1. Replace f(x) with y: y = 3x - 2 2. Swap x and y: x = 3y - 2 3. Solve for y: x + 2 = 3y y = (x + 2) / 3 The inverse of the function f(x) = 3x - 2 is f^(-1)(x) = (x + 2) / 3, which corresponds to option C. 4 / 5 4. Which of the following represents a system of equations that has no solution? a) y = 2x + 1 and y = 2x + 2 b) y = x - 3 and y = -x + 1 c) 3x + 6y = 9 and 2x - 3y = 4 d) y = 2x - 5 and y = -3x + 8 A system of equations has no solution if the two lines are parallel (meaning they have the same slope but different y-intercepts). Let's analyze the options: A) y = 2x + 1 and y = 2x + 2 These two lines have the same slope (2), and different y-intercepts (1 and 2). Thus, they are parallel and have no solution. B) y = x - 3 and y = -x + 1 These two lines have different slopes (1 and -1), so they are not parallel and will have a solution. C) 3x + 6y = 9 and 2x - 3y = 4 To check for parallel lines, put both equations in the slope-intercept form (y = mx + b). For the first equation: 6y = -3x + 9 y = -1/2x + 3/2 For the second equation: 3y = -2x + 4 y = -2/3x + 4/3 These two lines have different slopes (-1/2 and -2/3), so they are not parallel and will have a solution. D) y = 2x - 5 and y = -3x + 8 These two lines have different slopes (2 and -3), so they are not parallel and will have a solution. Thus, the system of equations with no solution is option A, y = 2x + 1 and y = 2x + 2. 5 / 5 5. If f(x) = 5xÂ² - 3x + 1 and g(x) = 2xÂ² + x - 3, what is f(x) - g(x)? a) 3xÂ² - 4x + 4 b) 3xÂ² + x + 2 c) 3xÂ² + 4x + 4 d) 7xÂ² - 2x + 4 To find the difference f(x) - g(x), we need to subtract the corresponding terms of f(x) and g(x): f(x) - g(x) = (5xÂ² - 3x + 1) - (2xÂ² + x - 3) Subtract g(x) from f(x): 3xÂ² - 4x + 4 Thus, f(x) - g(x) = 3xÂ² - 4x + 4, which corresponds to option A. Your score is 0% Restart Quiz