Question

The equations \(2x^2 + ax - 2 = 0\) and \(x^2 + x + 2a = 0\) have exactly one common root. If \(a \neq 0\), then one of the roots of the equation \(ax^2 - 4x - 2a = 0\) is:

• A. \(2\)
• B. \(-2\)
• C. \(-\frac{4 + \sqrt{22}}{3}\)
• D. \(-\frac{2 + \sqrt{22}}{3}\)

Hint:

Always verify that the derived roots satisfy all initial equations, particularly when they involve parameters like \(a\).

Answer 1

The Correct Option is D

Step 1: Identify the common root.
Assuming \(x_0\) is the common root for both equations, equate the two: \[ 2x_0^2 + ax_0 - 2 = 0 \quad \text{and} \quad x_0^2 + x_0 + 2a = 0. \] Subtract the second equation from the first (after adjusting coefficients), we derive an expression for \(a\) in terms of \(x_0\). Step 2: Substitute \(a\) into the third equation.
With the derived value of \(a\), substitute into \(ax^2 - 4x - 2a = 0\) and simplify to find the possible values of \(x\). Step 3: Solve for \(x\) in the modified third equation.
After substituting and simplifying, apply the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot a \cdot (-2a)}}{2a} \] Calculating further, we find: \[ x = -\frac{2 + \sqrt{22}}{3}. \]