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: Recall the sum and product of zeros for the quadratic ax^2 + bx + c = 0 . Sum of zeros: s = -(b)/(a) Product of zeros: p = (c)/(a) Given: s = (1)/(4) , p = -(1)/(4) s = -(b)/(a) = (1)/(4) p = (c)/(a) = -(1)/(4) Step 2: Choose a = 4 to obtain integer coefficients. Substitute a = 4 into sum formula: -(b)/(4) = (1)/(4) Multiply both sides by -4: b = -1 Step 3: Substitute a = 4 into product formula: (c)/(4) = -(1)/(4) Multiply both sides by 4: c = -1 Step 4: Form the quadratic polynomial. 4x^2 + (-1)x + (-1) = 4x^2 - x - 1 Step 5: Verify sum and product. Sum: -(b)/(a) = -(-1)/(4) = (1)/(4) Product: (c)/(a) = (-1)/(4) = -(1)/(4) 4x^2 - x - 1