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
Step 1: Factorize each expression. For the first expression, a^2-4: This is a difference of squares, x^2-y^2 = (x-y)(x+y). a^2-4 = (a-2)(a+2) For the second expression, 8a^2-14a-4: First, factor out the common numerical factor, which is 2. 8a^2-14a-4 = 2(4a^2-7a-2) Now, factor the quadratic 4a^2-7a-2. We look for two numbers that multiply to 4 × -2 = -8 and add to -7. These numbers are -8 and 1. 4a^2-7a-2 = 4a^2-8a+a-2 Factor by grouping: 4a(a-2)+1(a-2) = (4a+1)(a-2) So, the second expression is: 8a^2-14a-4 = 2(4a+1)(a-2) For the third expression, 12a^2-21a-6: First, factor out the common numerical factor, which is 3. 12a^2-21a-6 = 3(4a^2-7a-2) As determined above, 4a^2-7a-2 = (4a+1)(a-2). So, the third expression is: 12a^2-21a-6 = 3(4a+1)(a-2) Step 2: Determine the HCF (Highest Common Factor). The factored expressions are: 1. (a-2)(a+2) 2. 2(4a+1)(a-2) 3. 3(4a+1)(a-2) To find the HCF, we look for factors that are common to all three expressions. • The factor (a-2) is present in expressions 1 and 2, but not in expression 3. • The factor (a+2) is present only in expression 1. • The factor (4a+1) is present in expressions 2 and 3, but not in expression 1. • There are no common numerical factors across all three expressions (1, 2, and 3). Since there are no common factors other than 1 across all three expressions, the HCF is 1. HCF = 1 Step 3: Determine the LCM (Lowest Common Multiple). The LCM is the product of all unique factors (common and non-common) raised to the highest power they appear in any of the expressions. The unique factors are: • Numerical factors: 2 (from expression 2) and 3 (from expression 3). The highest power of 2 is 2^1. The highest power of 3 is 3^1. So, 2 × 3 = 6. • Algebraic factors: • (a-2): Appears in expressions 1 and 2. The highest power is 1. • (a+2): Appears in expression 1. The highest power is 1. • (4a+1): Appears in expressions 2 and 3. The highest power is 1. Multiply all these unique factors with their highest powers: LCM = 6(a-2)(a+2)(4a+1) The HCF and LCM are: HCF: 1 LCM: 6(a-2)(a+2)(4a+1) That's 2 down. 3 left today — send the next one.