Find two consecutive integers whose squares sum to 61.

Step 1: Define the two consecutive integers. Let the first integer be x. The next consecutive integer is x+1. Step 2: Set up the equation based on the given information. The sum of their squares is 61. x^2 + (x+1)^2 = 61 Step 3: Expand and simplify the equation. x^2 + (x^2 + 2x + 1) = 61 2x^2 + 2x + 1 = 61 Subtract 61 from both sides: 2x^2 + 2x - 60 = 0 Step 4: Divide the entire equation by 2. x^2 + x - 30 = 0 Step 5: Solve the quadratic equation by factoring. We need two numbers that multiply to -30 and add to 1. These numbers are 6 and -5. (x+6)(x-5) = 0 Step 6: Find the possible values for x. Set each factor to zero: x+6 = 0 x = -6 x-5 = 0 x = 5 Step 7: Determine the pairs of consecutive integers. If x = 5, the integers are 5 and 5+1 = 6. Check: 5^2 + 6^2 = 25 + 36 = 61. If x = -6, the integers are -6 and -6+1 = -5. Check: (-6)^2 + (-5)^2 = 36 + 25 = 61. The two integers are either 5 and 6 or -6 and -5. Step 1: Define the dimensions of the rectangle. Let the length of the rectangle be L cm. The width is 6 cm shorter than the length, so the width W = (L-6) cm. Step 2: Use the given area to form an equation. The area of a rectangle is given by A = L × W. Given A = 27 cm^2. 27 = L(L-6) Step 3: Expand and simplify the equation. 27 = L^2 - 6L Rearrange into a standard quadratic form: L^2 - 6L - 27 = 0 Step 4: Solve the quadratic equation for L. We need two numbers that multiply to -27 and add to -6. These numbers are -9 and 3. (L-9)(L+3) = 0 Step 5: Find the possible values for L. Set each factor to zero: L-9 = 0 L = 9 L+3 = 0 L = -3 Step 6: Choose the valid length. Since length cannot be negative, we take L = 9 cm. Step 7: Calculate the width. W = L - 6 = 9 cm - 6 cm = 3 cm Step 8: Calculate the perimeter of the rectangle. The perimeter of a rectangle is given by P = 2(L+W). P = 2(9 cm + 3 cm) P = 2(12 cm) P = 24 cm The perimeter of the rectangle is 24 cm. That's 2 down. 3 left today — send the next one.