If the equations $ 3x^2 - 2x +p =0 $ and $ 6x^2 -17x +12 = 0 $ have a common root , then the value of p is
- $\dfrac{15}{3} , \dfrac{8}{3}$
- $\dfrac{-15}{4} , \dfrac{-8}{3}$
- $\dfrac{15}{7} , \dfrac{8}{3}$
- $\dfrac{-15}{4} , \dfrac{8}{3}$
Roots of 6x^2 - 17x + 12 = 0 are (17 +/- sqrt(289 - 288))/12 = (17 +/- 1)/12. Roots are 18/12 = 3/2 and 16/12 = 4/3. If 3/2 is a common root: 3(9/4) - 2(3/2) + p = 0 => 27/4 - 3 + p = 0 => p = -15/4. If 4/3 is a common root: 3(16/9) - 2(4/3) + p = 0 => 16/3 - 8/3 + p = 0 => p = -8/3.
The roots of the quadratic equation 6x^2 - 17x + 12 = 0 are found using factorization to be 3/2 and 4/3. For a common root, we substitute these values into the first equation, 3x^2 - 2x + p = 0. Substituting x = 3/2 gives 3(9/4) - 2(3/2) + p = 0, which simplifies to 27/4 - 3 + p = 0, yielding p = -15/4. Substituting x = 4/3 gives 3(16/9) - 2(4/3) + p = 0, which simplifies to 16/3 - 8/3 + p = 0, yielding p = -8/3. The required values of p are -15/4 and -8/3.