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.
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Here is the solution to Question 9. Question 9: Given the quadratic equation 3x^2 - 11x + 4 = 0, with roots and . From the properties of quadratic equations, for ax^2 + bx + c = 0: Sum of roots: + = -(b)/(a) Product of roots: = (c)/(a) For 3x^2 - 11x + 4 = 0: + = -((-11))/(3) = (11)/(3) = (4)/(3) We need to find the equation whose roots are X_1 = ^2 + (1)/() and X_2 = (1)/() + ^2. A quadratic equation with roots X_1 and X_2 is given by x^2 - (X_1 + X_2)x + (X_1 X_2) = 0. Step 1: Calculate the sum of the new roots, S = X_1 + X_2. S = (^2 + (1)/()) + ((1)/() + ^2) S = ^2 + ^2 + (1)/() + (1)/() We know that ^2 + ^2 = ( + )^2 - 2 . And (1)/() + (1)/() = ( + )/( ). First, calculate ^2 + ^2: ^2 + ^2 = ((11)/(3))^2 - 2((4)/(3)) ^2 + ^2 = (121)/(9) - (8)/(3) ^2 + ^2 = (121)/(9) - (24)/(9) = (97)/(9) Next, calculate (1)/() + (1)/(): (1)/() + (1)/() = (11/3)/(4/3) = (11)/(4) Now, substitute these values back into the expression for S: S = (97)/(9) + (11)/(4) To add these fractions, find a common denominator, which is 36. S = (97 × 4)/(9 × 4) + (11 × 9)/(4 × 9) S = (388)/(36) + (99)/(36) = (487)/(36) Step 2: Calculate the product of the new roots, P = X_1 X_2. P = (^2 + (1)/()) ((1)/() + ^2) Expand the product: P = ^2 · (1)/() + ^2 ^2 + (1)/() · (1)/() + (1)/() ^2 P = + ( )^2 + (1)/( ) + Rearrange the terms: P = ( + ) + ( )^2 + (1)/( ) Substitute the values of + and : P = (11)/(3) + ((4)/(3))^2 + (1)/(4/3) P = (11)/(3) + (16)/(9) + (3)/(4) To add these fractions, find a common denominator, which is 36. P = (11 × 12)/(3 × 12) + (16 × 4)/(9 × 4) + (3 × 9)/(4 × 9) P = (132)/(36) + (64)/(36) + (27)/(36) P = (132 + 64 + 27)/(36) = (223)/(36) Step 3: Form the new quadratic equation x^2 - Sx + P = 0. Substitute the calculated values of S and P: x^2 - (487)/(36)x + (223)/(36) = 0 Multiply the entire equation by 36 to eliminate the denominators: 36x^2 - 487x + 223 = 0 36x^2 - 487x + 223 = 0 Last free one today — make it count tomorrow, or type /upgrade for unlimited.