Solve both equations and form a new equation in variable 'z' (reduce to lowest…
2021
Solve both equations and form a new equation in variable 'z' (reduce to lowest possible factor) using roots of equation 1 and 2 as per instructions given below.
2 - 11/x + 9/x² = 0
(y - 2)² = 2 ¼
What will be new equation if roots of this are highest root of equation 1 and lowest root of equation 2.
- A.
8z² - 34z - 9 = 0
- B.
4z² - 20z + 9 = 0
- C.
8z² - 20z + 9 = 0
- D.
4z² - 34z - 9 = 0
- E.
None of these
Attempted by 2 students.
Show answer & explanation
Correct answer: B
Concept
If a quadratic has roots p and q, it can be written as z² − (p + q)z + (p·q) = 0, i.e. the coefficient of z is the negative of the sum of roots and the constant term is the product of roots. So to build the target equation you only need the sum and product of the two chosen roots; clearing fractions then scales every term to integers.
Application
Step through each equation:
Equation 1: multiply 2 − 11/x + 9/x² = 0 by x² to get 2x² − 11x + 9 = 0. Discriminant = (−11)² − 4·2·9 = 121 − 72 = 49, so x = (11 ± 7)/4, giving x = 18/4 = 4.5 and x = 4/4 = 1. Highest root of equation 1 = 4.5.
Equation 2: (y − 2)² = 2¼ = 9/4, so y − 2 = ±3/2, giving y = 3.5 and y = 0.5. Lowest root of equation 2 = 0.5.
Chosen roots are 4.5 and 0.5. Sum = 4.5 + 0.5 = 5; Product = 4.5 × 0.5 = 2.25.
Form the equation: z² − 5z + 2.25 = 0. Multiply through by 4 to clear the decimal: 4z² − 20z + 9 = 0.
Cross-check
Substitute the roots into 4z² − 20z + 9 = 0. For z = 4.5: 4(20.25) − 20(4.5) + 9 = 81 − 90 + 9 = 0. For z = 0.5: 4(0.25) − 20(0.5) + 9 = 1 − 10 + 9 = 0. Both satisfy it, and the leading coefficient 4 is the smallest that makes all coefficients integers, so the equation is in lowest form.