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
For question 21, your factorization and solutions are correct. For question 22, your derivation of the quadratic equation 12x^2 - 5x - 3 = 0 is correct. For question 23, your setup of the quadratic equation 2x^2 - 51x + 115 = 0 is correct. Let's complete the solution from there. Step 1: Solve the quadratic equation for x using factorization. The equation is 2x^2 - 51x + 115 = 0. We need two numbers that multiply to 2 × 115 = 230 and add up to -51. These numbers are -5 and -46. Rewrite the middle term: 2x^2 - 5x - 46x + 115 = 0 Step 2: Factor by grouping. x(2x - 5) - 23(2x - 5) = 0 (x - 23)(2x - 5) = 0 Step 3: Find the possible values for x. x - 23 = 0 or 2x - 5 = 0 x = 23 or 2x = 5 x = 23 or x = (5)/(2) = 2.5 Step 4: Check the validity of the x values. The width of the margin x must be less than half of the smallest dimension of the frame. The frame width is 21 cm, so 2x must be less than 21 cm, meaning x < 10.5 cm. If x = 23 cm, then 2x = 46 cm. This would make the picture width 21 - 46 = -25 cm, which is not possible. So x = 23 is not a valid solution. If x = 2.5 cm, then 2x = 5 cm. This is a valid margin width. Step 5: Calculate the dimensions of the picture using the valid value of x. Picture length = 30 - 2x = 30 - 2(2.5) = 30 - 5 = 25 cm Picture width = 21 - 2x = 21 - 2(2.5) = 21 - 5 = 16 cm The dimensions of the picture are 25 cm by 16 cm. The final answer is Length = 25 cm, Width = 16 cm. 3 done, 2 left today. You're making progress.