Question

If the roots of the quadratic equation \( 2x^2 - 4x + k = 0 \) are real and equal, find the value of \( k \).

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

For equal roots, set the discriminant \( b^2 - 4ac = 0 \) and solve for the unknown coefficient, then verify by checking the roots.

Answer 1

The Correct Option is A

For a quadratic equation \( ax^2 + bx + c = 0 \), the roots are real and equal if the discriminant is zero: \[ \Delta = b^2 - 4ac = 0 \] Here, \( a = 2 \), \( b = -4 \), \( c = k \). Thus: \[ (-4)^2 - 4 \cdot 2 \cdot k = 0 \] \[ 16 - 8k = 0 \implies 8k = 16 \implies k = 2 \] Verify: \[ 2x^2 - 4x + 2 = 0 \implies x^2 - 2x + 1 = 0 \implies (x - 1)^2 = 0 \] Roots are \( x = 1, 1 \), which are real and equal. Thus, the value of \( k \) is: \[ \boxed{2} \]