Multiple choice

If one root of the quadratic equation $3x^{2}-10x+p=0$ is $\dfrac{1}{3},$ then the value of $p$ and the other root respectively is :

  1. $\displaystyle 3,\frac{1}{3}$
  2. $\displaystyle 3,3$
  3. $\displaystyle -\frac{1}{3},-\frac{1}{3}$
  4. $\displaystyle -3,-3$
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B Correct answer
Explanation

If 1/3 is a root, 3(1/9) - 10(1/3) + p = 0. 1/3 - 10/3 + p = 0. -9/3 + p = 0. p = 3. Equation is 3x^2 - 10x + 3 = 0. (3x - 1)(x - 3) = 0. Roots are 1/3 and 3.

AI explanation

For the quadratic equation 3x^2 - 10x + p = 0, the sum of the roots is given by -(-10)/3 = 10/3. Since one root is the reciprocal of the leading coefficient, 1/3, we subtract it from the sum to find the other root: 10/3 - 1/3 = 9/3 = 3. The product of the roots gives p/3, so multiplying the two roots yields p/3 = 1/3 * 3 = 1. Therefore, p equals 3, and the other root is 3.

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