Question

If one root of the quadratic equation \(2x^2 - x - 6 = 0\) is \(-\frac{3}{2}\) then its another root is

• A. -2
• B. 2
• C. \(\frac{3}{2}\)
• D. 3

Hint:

Using the sum of roots is generally simpler than the product when finding the second root, as it involves addition/subtraction rather than division. However, using the product is a great way to double-check your answer.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a quadratic equation \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\), we know the sum of roots is \(\alpha + \beta = -b/a\) and the product is \(\alpha\beta = c/a\). If one root is known, we can use either of these relations to find the other.

Step 2: Key Formula or Approach:
Using the sum of roots is often the easiest. Let the known root be \(\alpha\) and the unknown root be \(\beta\).
\[ \alpha + \beta = -\frac{b}{a} \]

Step 3: Detailed Explanation:
The equation is \(2x^2 - x - 6 = 0\), so \(a=2, b=-1, c=-6\).
One root is given as \(\alpha = -\frac{3}{2}\).
Using the sum of roots formula:
\[ -\frac{3}{2} + \beta = -\frac{-1}{2} \] \[ -\frac{3}{2} + \beta = \frac{1}{2} \] \[ \beta = \frac{1}{2} + \frac{3}{2} \] \[ \beta = \frac{4}{2} = 2 \] To verify, let's use the product of roots: \(\alpha\beta = \frac{c}{a} = \frac{-6}{2} = -3\).
\[ \left(-\frac{3}{2}\right) \times 2 = -3 \] The product matches, so the calculation is correct.

Step 4: Final Answer:
The another root is 2.