This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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The question "5) The Sum of two numbers is and their product is 77." is incomplete as the value for the sum is missing. I will assume the sum of the two numbers is 18, as this is a common value that leads to integer solutions for the given product. a) Form a quadratic equation whose roots are the numbers Step 1: Define variables and state the assumed sum and given product. Let the two numbers be x_1 and x_2. We are given the product: x_1 · x_2 = 77. We assume the sum: x_1 + x_2 = 18. Step 2: Use the relationship between roots and coefficients to form the quadratic equation. A quadratic equation with roots x_1 and x_2 can be written in the form: x^2 - (x_1 + x_2)x + (x_1 · x_2) = 0 Substitute the sum and product into the equation. x^2 - (18)x + (77) = 0 The quadratic equation is x^2 - 18x + 77 = 0. b) Solve the equation Step 3: Solve the quadratic equation by factoring. We need to find two numbers that multiply to 77 and add up to -18. These numbers are -7 and -11. (x - 7)(x - 11) = 0 Set each factor to zero to find the roots. x - 7 = 0 or x - 11 = 0 x = 7 or x = 11 The solutions to the equation are x = 7 and x = 11. c) Hence find the two numbers Step 4: Identify the two numbers from the solutions of the quadratic equation. The roots of the quadratic equation are the two numbers. Therefore, the two numbers are 7 and 11. That's 2 down. 3 left today — send the next one.