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.
ScanSolve AI Answer
Step 1: Define the consecutive odd integers. Let the first odd integer be x. Since the integers are consecutive and odd, the next odd integer will be x+2. Step 2: Formulate the equation based on the given information. The product of the two consecutive odd integers is x(x+2). Their sum is x + (x+2) = 2x+2. Four times their sum is 4(2x+2). The problem states that the product is 1 less than four times their sum. So, the equation is: x(x+2) = 4(2x+2) - 1 Step 3: Solve the quadratic equation for x. Expand both sides of the equation: x^2 + 2x = 8x + 8 - 1 x^2 + 2x = 8x + 7 Rearrange the terms to form a standard quadratic equation ax^2 + bx + c = 0: x^2 + 2x - 8x - 7 = 0 x^2 - 6x - 7 = 0 Factor the quadratic equation: We need two numbers that multiply to -7 and add to -6. These numbers are -7 and 1. (x-7)(x+1) = 0 Set each factor to zero to find the possible values for x: x-7 = 0 x = 7 x+1 = 0 x = -1 Step 4: Find the pairs of consecutive odd integers. Case 1: If x = 7 The first integer is 7. The second integer is x+2 = 7+2 = 9. The pair of integers is (7, 9). Case 2: If x = -1 The first integer is -1. The second integer is x+2 = -1+2 = 1. The pair of integers is (-1, 1). Both pairs satisfy the given condition. The integers are 7 and 9 or -1 and 1. 3 done, 2 left today. You're making progress.