Form a quadratic equation in x to represent the information and find the value of x.

Here are the solutions to the problems. Problem 1: a) Form a Quadratic equation in x to represent the information. Step 1: Define variables and set up the initial equation. Let x be the original number of women. The total amount to be contributed is Sh. 1200. The original contribution per woman would be (1200)/(x). 5 women were unable to contribute, so the number of contributing women became x-5. The new contribution per woman is (1200)/(x-5). The problem states that the remaining women contributed Sh. 20 more each. (1200)/(x-5) = (1200)/(x) + 20 Step 2: Rearrange the equation to isolate the constant term. (1200)/(x-5) - (1200)/(x) = 20 Step 3: Combine the fractions on the left side using a common denominator x(x-5). (1200x - 1200(x-5))/(x(x-5)) = 20 (1200x - 1200x + 6000)/(x^2 - 5x) = 20 (6000)/(x^2 - 5x) = 20 Step 4: Multiply both sides by (x^2 - 5x). 6000 = 20(x^2 - 5x) Step 5: Divide both sides by 20. (6000)/(20) = x^2 - 5x 300 = x^2 - 5x Step 6: Rearrange into the standard quadratic form ax^2 + bx + c = 0. x^2 - 5x - 300 = 0 The quadratic equation is x^2 - 5x - 300 = 0. b) Find the value of x. Step 1: Solve the quadratic equation x^2 - 5x - 300 = 0 by factoring. We need two numbers that multiply to -300 and add to -5. These numbers are -20 and 15. (x - 20)(x + 15) = 0 Step 2: Determine the possible values for x. x - 20 = 0 x = 20 x + 15 = 0 x = -15 Step 3: Choose the valid value for x. Since x represents the number of women, it must be a positive value. Therefore, x = 20. The value of x is 20. c) Determine the number of women that contributed the money and the amount each contributed. Step 1: Calculate the number of women who contributed. The original number of women was x = 20. 5 women were unable to contribute, so the number of contributing women is x - 5. Number of contributing women = 20 - 5 = 15 women. Step 2: Calculate the amount each contributing woman paid. The total amount contributed was Sh. 1200. Amount each contributed = Total amountNumber of contributing women Amount each contributed = (1200)/(15) = 80 Shillings. The number of women that contributed the money is 15 and the amount each contributed is Sh. 80. Problem 2: Write an expression for the area of a triangular ranch in terms of x. Step 1: Recall the formula for the area of a triangle. A = (1)/(2) × base × height Step 2: Substitute the given base and height into the formula. Base = (x+14) km Height = (x-2) km A = (1)/(2)(x+14)(x-2) Step 3: Expand the product of the binomials. (x+14)(x-2) = x(x-2) + 14(x-2) = x^2 - 2x + 14x - 28 = x^2 + 12x - 28 Step 4: Substitute the expanded expression back into the area formula and simplify. A = (1)/(2)(x^2 + 12x - 28) A = (1)/(2)x^2 + (12)/(2)x - (28)/(2) A = (1)/(2)x^2 + 6x - 14 The expression for the area is (1)/(2)x^2 + 6x - 14 km^2. Send me the next one 📸