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: Rearrange the terms of the quadratic expression into standard form ax^2 + bx + c. The given expression is 2 + 5d - 12d^2. Rearranging it gives: -12d^2 + 5d + 2 Step 2: Factor out -1 to make the leading coefficient positive, or factor directly. Let's factor directly by finding two numbers that multiply to a × c = (-12) × 2 = -24 and add up to b = 5. The two numbers are 8 and -3. Step 3: Rewrite the middle term 5d using these two numbers. -12d^2 + 8d - 3d + 2 Step 4: Group the terms and factor out common factors from each group. (-12d^2 + 8d) + (-3d + 2) Factor 4d from the first group and -1 from the second group: 4d(-3d + 2) - 1(3d - 2) To make the binomials identical, we can factor out -4d from the first group instead: -4d(3d - 2) - 1(3d - 2) Step 5: Factor out the common binomial (3d - 2). (3d - 2)(-4d - 1) This can also be written as (2 - 3d)(4d + 1) by factoring out -1 from the first factor and -1 from the second factor, then multiplying the two -1s to get +1, or by factoring -(12d^2 - 5d - 2). Let's verify the factorization: (3d - 2)(-4d - 1) = 3d(-4d) + 3d(-1) - 2(-4d) - 2(-1) = -12d^2 - 3d + 8d + 2 = -12d^2 + 5d + 2 This matches the original expression. The completely factorised expression is (3d - 2)(-4d - 1). That's 2 down. 3 left today — send the next one.