If x₁ and x₂ are the roots of the equation x² + 2x - 15 = 0 then the quadratic…
2017
If x₁ and x₂ are the roots of the equation x² + 2x - 15 = 0 then the quadratic equation which has the roots 1/x₁ and 1/x₂ is:
- A.
15x² - 2x - 1 = 0
- B.
15x² + 2x - 1 = 0
- C.
-15x² - 2x - 1 = 0
- D.
15x² - 2x + 1 = 0
Attempted by 3 students.
Show answer & explanation
Correct answer: A
Concept
For a quadratic ax² + bx + c = 0 with roots p and q, the sum of roots is p + q = -b/a and the product is pq = c/a. To obtain the equation whose roots are the reciprocals 1/p and 1/q, note that their sum is (p + q)/(pq) and their product is 1/(pq); the required equation is then x² - (new sum)x + (new product) = 0.
Application
From x² + 2x - 15 = 0, identify a = 1, b = 2, c = -15, so x₁ + x₂ = -b/a = -2 and x₁·x₂ = c/a = -15.
Sum of the new roots: 1/x₁ + 1/x₂ = (x₁ + x₂)/(x₁·x₂) = (-2)/(-15) = 2/15.
Product of the new roots: (1/x₁)(1/x₂) = 1/(x₁·x₂) = 1/(-15) = -1/15.
Form the equation: x² - (2/15)x + (-1/15) = 0.
Clear the denominator by multiplying every term by 15: 15x² - 2x - 1 = 0.
Cross-check
Factor the original equation: x² + 2x - 15 = (x + 5)(x - 3) = 0, so the roots are x = -5 and x = 3. The reciprocals are -1/5 and 1/3. Their sum is -1/5 + 1/3 = 2/15 and their product is -1/15, which match the coefficients of 15x² - 2x - 1 = 0. Substituting x = 1/3 gives 15(1/9) - 2(1/3) - 1 = 5/3 - 2/3 - 1 = 0, confirming the result.