Multiple choice

If the equations 13x2 - 7x - 6 = 0 and 2x2 + 5x + k = 0 have exactly one root in common, then the values of k can be

  1. (- 7) or (- 38/3)

  2. (7) or (- 38/3)

  3. (- 7) or (38/3)

  4. (7) or (38/3)

  5. ( 7) or (-7)
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AI explanation

By the quadratic formula, the roots of 13x2 - 7x - 6 = 0 are calculated as (7 ± 49 + 312) / 26, which simplifies to (7 ± 19) / 26, giving the roots 1 and -6/13. Substituting x = 1 into the second equation 2x2 + 5x + k = 0 yields 2(1) + 5(1) + k = 0, so k = -7. Substituting x = -6/13 into the second equation gives 2(36/169) + 5(-6/13) + k = 0, which simplifies to 72/169 - 390/169 + k = 0, meaning k = 318/169 = -38/3. The values of k are -7 or -38/3.

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