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: Identify the sum and product of roots for the original equation. The given equation is x^2 + px + q = 0. Let its roots be and . Using Vieta's formulas: Sum of roots: + = -p Product of roots: = q Step 2: Calculate the sum of the new roots. The new roots are (1)/(^2) and (1)/(^2). Let S' be the sum of the new roots: S' = (1)/(^2) + (1)/(^2) Combine the fractions: S' = (^2 + ^2)/(^2 ^2) We know that ^2 + ^2 = ( + )^2 - 2. Substitute the values from Step 1: ^2 + ^2 = (-p)^2 - 2(q) = p^2 - 2q Also, ^2 ^2 = ()^2 = q^2. Substitute these into the expression for S': S' = (p^2 - 2q)/(q^2) Step 3: Calculate the product of the new roots. Let P' be the product of the new roots: P' = ((1)/(^2)) ((1)/(^2)) P' = (1)/(^2 ^2) Substitute the value from Step 1: P' = (1)/(()^2) = (1)/(q^2) Step 4: Form the new quadratic equation. A quadratic equation with roots r_1 and r_2 can be written as x^2 - (r_1 + r_2)x + r_1 r_2 = 0, or x^2 - S'x + P' = 0. Substitute the calculated values for S' and P': x^2 - ((p^2 - 2q)/(q^2))x + (1)/(q^2) = 0 To eliminate the denominators, multiply the entire equation by q^2: q^2 x^2 - (p^2 - 2q)x + 1 = 0 The new quadratic equation is q^2 x^2 - (p^2 - 2q)x + 1 = 0. That's 2 down. 3 left today — send the next one.