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
I will solve question 7(e) from the image. The question asks to factorize the expression a^2 + 5ab - 84b^2. Step 1: Identify the coefficients. This is a quadratic expression in terms of a and b. We need to find two terms that multiply to the product of the first and last terms (a^2 × -84b^2 = -84a^2b^2) and add up to the middle term (5ab). Step 2: Find two numbers that multiply to -84 and add to 5. We need two numbers whose product is -84 and whose sum is 5. These numbers are 12 and -7. So, the terms are 12ab and -7ab. Step 3: Rewrite the middle term using these two terms. a^2 + 5ab - 84b^2 = a^2 + 12ab - 7ab - 84b^2 Step 4: Factor by grouping. Group the terms and factor out the common factors from each group. (a^2 + 12ab) - (7ab + 84b^2) Factor a from the first group and 7b from the second group. a(a + 12b) - 7b(a + 12b) Step 5: Factor out the common binomial. (a + 12b)(a - 7b) The factored form of a^2 + 5ab - 84b^2 is (a + 12b)(a - 7b). The final answer is (a + 12b)(a - 7b). 3 done, 2 left today. You're making progress.