In a cubic equation , where its roots are :
- Sum of the roots is therefore
- Product of the roots is therefore
- Sum of the roots taken two at a time is therefore
Combine that with this. Harmonic Mean = or Harmonic Mean = . Whichever form works better for the given problem.
HARMONIC MEAN
HARMONIC MEAN
HARMONIC MEAN
Practice:
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is -9 and PQ+QR+PR is -16. Find PQR.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is and PQR is -20. Find C.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is 6 and PQR is -72. Find C.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is 9 and PQ+PR+QR is 4. Find PQR.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is -7 and PQ+PR+QR is -22. Find D.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is -3 and PQR is 15. Find PQ+PR+QR.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is 10 and D is 168. Find PQ+PR+QR.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is -3 and C is 36. Find D.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is 7 and C is -10. Find PQR.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is -3 and D is -28. Find C.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is 2 and PQR is 49. Find C.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is -2 and PQ+PR+QR is -3. Find PQR.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is -4 and PQ+PR+QR is -9. Find D.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is 3 and PQR is -16. Find PQ+PR+QR.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is -3 and D is 63. Find PQ+PR+QR.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is -8 and C is -33. Find D.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is -4 and C is 54. Find PQR.
P, Q, and R are the real roots of . The harmonic mean of P, Q, and R is 15 and D is 20. Find C.
Answers: 48?, 11?, -36, 12, -56, -13, -46, 36, -24, -25, 63, 2, -14, -16, 51, -90, -72, -4.