Question

If \(\alpha\) be a root of the equation \(4x^2+2x-1=0\), then the other root is

• A. \(-2\alpha-1\)
• B. \(4\alpha^2+\alpha-1\)
• C. \(4\alpha^3-3\alpha\)
• D. \(4\alpha^2-3\alpha\)

Hint:

For any quadratic: \[ \text{Other root} = \left(-\frac{b}{a}\right)-\text{given root} \] Always use the sum of roots relation to express one root in terms of the other.

Answer 1

The Correct Option is A

Step 1: For a quadratic equation: \[ ax^2+bx+c=0 \] If the roots are \(\alpha\) and \(\beta\), then: \[ \alpha+\beta=-\frac{b}{a}, \qquad \alpha\beta=\frac{c}{a} \]
Step 2: Given equation: \[ 4x^2+2x-1=0 \] Here, \[ a=4,\quad b=2,\quad c=-1 \]
Step 3: Sum of roots: \[ \alpha+\beta=-\frac{2}{4}=-\frac{1}{2} \]
Step 4: Express the other root \(\beta\) in terms of \(\alpha\): \[ \beta=-\frac{1}{2}-\alpha \]
Step 5: Rewrite \(\beta\): \[ \beta=-\frac{1+2\alpha}{2}=-2\alpha-1 \quad (\text{equivalent form}) \]
Step 6: Hence, the other root is: \[ \boxed{-2\alpha-1} \]