Find the slope of the given line.

Step 1: Find the slope of the given line. The equation of the given line is 3y = 5 - 2x. To find its slope, rewrite it in the slope-intercept form y = mx + c: 3y = -2x + 5 y = -(2)/(3)x + (5)/(3) The slope of this line, let's call it m_1, is -(2)/(3). Step 2: Find the slope of the line passing through A and B. The line passing through points A (2, 1) and B (4, m) is perpendicular to the given line. If two lines are perpendicular, the product of their slopes is -1. Let m_2 be the slope of the line through A and B. m_1 × m_2 = -1 -(2)/(3) × m_2 = -1 m_2 = (-1)/(-2)3 m_2 = (3)/(2) Step 3: Use the slope formula to find the value of m. The slope of a line passing through two points (x_1, y_1) and (x_2, y_2) is given by m = (y_2 - y_1)/(x_2 - x_1). Using points A (2, 1) and B (4, m): m_2 = (m - 1)/(4 - 2) (3)/(2) = (m - 1)/(2) Multiply both sides by 2: 3 = m - 1 Add 1 to both sides: m = 3 + 1 m = 4 The value of m is 4. Step 1: Divide the equation by the coefficient of x^2. The given quadratic equation is 2x^2 - 5x + 3 = 0. Divide all terms by 2: x^2 - (5)/(2)x + (3)/(2) = 0 Step 2: Move the constant term to the right side. x^2 - (5)/(2)x = -(3)/(2) Step 3: Complete the square on the left side. To complete the square for x^2 + bx, add ((b)/(2))^2 to both sides. Here, b = -(5)/(2). ((b)/(2))^2 = ((-5)/(2)2)^2 = (-(5)/(4))^2 = (25)/(16) Add (25)/(16) to both sides of the equation: x^2 - (5)/(2)x + (25)/(16) = -(3)/(2) + (25)/(16) Step 4: Factor the left side and simplify the right side. The left side is a perfect square: (x - (5)/(4))^2. For the right side, find a common denominator: -(3)/(2) + (25)/(16) = -(3 × 8)/(2 × 8) + (25)/(16) = -(24)/(16) + (25)/(16) = (1)/(16) So the equation becomes: (x - (5)/(4))^2 = (1)/(16) Step 5: Take the square root of both sides. x - (5)/(4) = ±sqrt((1)/(16)) x - (5)/(4) = ±(1)/(4) Step 6: Solve for x. x = (5)/(4) ± (1)/(4) Two possible solutions: x_1 = (5)/(4) + (1)/(4) = (6)/(4) = (3)/(2) x_2 = (5)/(4) - (1)/(4) = (4)/(4) = 1 The solutions are x = (3)/(2) or x = 1. Step 1: Apply logarithm properties to simplify the equation. The given equation is Log (x-1) = log 12 - log (x-2). Use the logarithm property log a - log b = log ((a)/(b)) on the right side: Log (x-1) = log ((12)/(x-2)) Step 2: Equate the arguments of the logarithms. Since the logarithms on both sides have the same base (implied base 10), their arguments must be equal: x-1 = (12)/(x-2) Step 3: Solve the resulting algebraic equation. Multiply both sides by (x-2) to eliminate the denominator: (x-1)(x-2) = 12 Expand the left side: x^2 - 2x - x + 2 = 12 x^2 - 3x + 2 = 12 Subtract 12 from both sides to set the quadratic equation to zero: x^2 - 3x - 10 = 0 Factor the quadratic equation: We need two numbers that multiply to -10 and add to -3. These numbers are -5 and 2. (x-5)(x+2) = 0 This gives two possible solutions for x: x-5 = 0 x = 5 x+2 = 0 x = -2 Step 4: Check for valid solutions. For a logarithm log A to be defined, the argument A must be positive (A > 0). Check x=5: x-1 = 5-1 = 4 > 0 (Valid) x-2 = 5-2 = 3 > 0 (Valid) So, x=5 is a valid solution. Check x=-2: x-1 = -2-1 = -3 (Not valid, as argument must be positive) x-2 = -2-2 = -4 (Not valid) So, x=-2 is not a valid solution. The only valid solution is x=5. 3 done, 2 left today. You're making progress.