Math Quiz -Algebra Report a question What's wrong with this question? You cannot submit an empty report. Please add some details. Math Quiz -Algebra Covering: linear equations, inequalities, systems of equations, absolute value expressions, and quadratic equations. 1 / 5 1. Find the slope of the line that passes through the points (1, 2) and (3, 5). a) 3/2 b) 2/3 c) -1.5 d) -0.66666666666667 To find the slope of the line that passes through points (1, 2) and (3, 5), we can use the slope formula: m = (y2 - y1) / (x2 - x1) Using the given points, the slope formula becomes: m = (5 - 2) / (3 - 1) = 3 / 2 Thus, the slope of the line is 3/2, which corresponds to option A. 2 / 5 2. For which value of x is the function f(x) = x³ - x² - 6x equal to 0? a) x = -2 b) x = 3 c) x = 0 d) x = 6 To solve the equation f(x) = x³ - x² - 6x = 0, we can factor out a common x term: x(x² - x - 6) = 0 Now we further factor the quadratic expression: x(x - 3)(x + 2) = 0 Setting each factor equal to zero yields the solutions: x = 0, x = 3, and x = -2 The function f(x) = x³ - x² - 6x is equal to 0 when x = 0, x = 3, and x = -2. The question asked for one value, so we choose x = 3, which corresponds to option B. 3 / 5 3. Solve the following system of linear equations: 2x + 3y = 6 and 4x - y = 5. a) x = -21/10, y = 17/5 b) x = 21/10, y = -17/5 c) x = -21/10, y = -17/5 d) x = 21/10, y = 17/5 To solve the system of linear equations, we can either use the substitution method or the elimination method. In this case, we'll use the elimination method. Note that 2x + 3y = 6 (equation 1) and 4x - y = 5 (equation 2). Step 1: Multiply equation 1 by 2 and add it to equation 2 to eliminate x: 4x + 6y = 12 (equation 1 multiplied by 2) + 4x - y = 5 (equation 2) Result: 5y = 17 Divide by 5: y = 17/5 Step 2: Substitute the value of y in the first equation: 2x + 3(17/5) = 6 2x + 51/5 = 6 Multiply by 5 to eliminate fractions: 10x + 51 = 30 Subtract 51 from both sides: 10x = -21 Divide by 10: x = -21/10 Thus, the solution to the system of linear equations is x = -21/10 and y = 17/5, which corresponds to option B. 4 / 5 4. Solve the quadratic inequality x² - 6x + 8 < 0. a) x < 2, x > 4 b) 2 < x < 4 c) x ≤ 2, x ≥ 4 d) 2 ≤ x ≤ 4 To solve the quadratic inequality x² - 6x + 8 < 0, we first need to factor the quadratic expression: x² - 6x + 8 = (x - 2)(x - 4) Now we find the critical points by setting each factor equal to zero: x - 2 = 0 => x = 2 x - 4 = 0 => x = 4 We'll test the intervals (-∞, 2), (2, 4), and (4, ∞) to see if the inequality is true: Interval (-∞, 2): Test x = 1 (1 - 2)(1 - 4) = 3 > 0 (not true) Interval (2, 4): Test x = 3 (3 - 2)(3 - 4) = -1 < 0 (true) Interval (4, ∞): Test x = 5 (5 - 2)(5 - 4) = 3 > 0 (not true) Thus, the solution to the quadratic inequality x² - 6x + 8 < 0 is 2 < x < 4, which corresponds to option B. 5 / 5 5. Using the quadratic formula, determine if the quadratic equation 2x² + 2x - 1 = 0 has real, imaginary, or no solutions. a) Two real solutions b) Two imaginary solutions c) One real solution d) No solutions The discriminant (∆) in the quadratic formula helps us determine the nature of the solutions for a quadratic equation ax² + bx + c = 0: ∆ = b² - 4ac If ∆ > 0, there are two real solutions. If ∆ = 0, there is exactly one real solution. If ∆ < 0, there are two imaginary (complex) solutions. For the given quadratic equation, we have: a = 2, b = 2, and c = -1 Now calculate the discriminant: ∆ = (2)² - 4(2)(-1) ∆ = 4 + 8 ∆ = 12 Since ∆ > 0, the quadratic equation has two real solutions, which corresponds to option A. Your score is 0% Restart Quiz