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 is the solution to Question 19. Question 19: Given that 9x^2 - 30x - 40 + k is a perfect square, find the value of k. Step 1: Recall the form of a perfect square trinomial. A perfect square trinomial is of the form (ax ± b)^2 = a^2x^2 ± 2abx + b^2. Comparing 9x^2 - 30x - 40 + k with this form, we can identify a^2x^2 = 9x^2 and 2abx = -30x. Step 2: Determine the values of a and b. From a^2x^2 = 9x^2, we get a = 3. From 2abx = -30x, substitute a=3: 2(3)bx = -30x 6bx = -30x 6b = -30 b = -5 (or we can consider b=5 and the form (ax-b)^2) Step 3: Determine the constant term of the perfect square. If a=3 and b=5, the perfect square is (3x - 5)^2. Expanding (3x - 5)^2: (3x - 5)^2 = (3x)^2 - 2(3x)(5) + 5^2 = 9x^2 - 30x + 25 Step 4: Equate the constant term of the given expression to the constant term of the perfect square. The given expression is 9x^2 - 30x - 40 + k. For this to be a perfect square, its constant term must be 25. So, we set -40 + k = 25. Step 5: Solve for k. -40 + k = 25 k = 25 + 40 k = 65