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
Here are the solutions to questions 49 and 50. 49. The expression x^2 + 10x + 18 can be written as (x+p)^2 + q. Find the values of p and q. Step 1: Expand the given form (x+p)^2 + q. (x+p)^2 + q = x^2 + 2px + p^2 + q Step 2: Compare the expanded form with the given expression x^2 + 10x + 18. By comparing the coefficients of x: 2p = 10 p = (10)/(2) p = 5 Step 3: Compare the constant terms. p^2 + q = 18 Substitute the value of p=5 into this equation: (5)^2 + q = 18 25 + q = 18 q = 18 - 25 q = -7 The values are p=5 and q=-7. This corresponds to option C. The final answer is C. 5 and -7. 50. Solve: 2x - 3 ≤ 4x + 5 < 3x + 7. This is a compound inequality that can be split into two separate inequalities. Step 1: Solve the first inequality: 2x - 3 ≤ 4x + 5. Subtract 2x from both sides: -3 ≤ 2x + 5 Subtract 5 from both sides: -3 - 5 ≤ 2x -8 ≤ 2x Divide by 2: (-8)/(2) ≤ x -4 ≤ x Step 2: Solve the second inequality: 4x + 5 < 3x + 7. Subtract 3x from both sides: x + 5 < 7 Subtract 5 from both sides: x < 7 - 5 x < 2 Step 3: Combine the solutions from Step 1 and Step 2. We have -4 ≤ x and x < 2. Combining these gives: -4 ≤ x < 2 This corresponds to option B. The final answer is B. -4 ≤ x < 2. 3 done, 2 left today. You're making progress.