Question

For what value of \( k \), the roots of the quadratic equation \( x^2 + 6x + k = 0 \) are real and equal?

• A. 12
• B. 9
• C. 10
• D. 6

Hint:

For real and equal roots of a quadratic equation, set the discriminant \( \Delta = 0 \) and solve for \( k \).

Answer 1

The Correct Option is B

For a quadratic equation \( ax^2 + bx + c = 0 \), the roots are real and equal if the discriminant \( \Delta \) is zero. The discriminant \( \Delta \) is given by: \[ \Delta = b^2 - 4ac. \] For the quadratic equation \( x^2 + 6x + k = 0 \), \( a = 1 \), \( b = 6 \), and \( c = k \). The discriminant is: \[ \Delta = 6^2 - 4(1)(k) = 36 - 4k. \] For real and equal roots, \( \Delta = 0 \), so: \[ 36 - 4k = 0 \quad \Rightarrow \quad 4k = 36 \quad \Rightarrow \quad k = 9. \] Thus, the value of \( k \) is \( \boxed{9} \).