Question

If the roots of the quadratic equation \( x^2 - 6x + k = 0 \) have a difference of 2, find the value of \( k \).

• A. 5
• B. 7
• C. 8
• D. 9

Hint:

For a quadratic equation, use the difference of roots formula \( |p - q| = \sqrt{(p + q)^2 - 4pq} \) to relate the sum and product of roots.

Answer 1

The Correct Option is C

For the quadratic equation \( x^2 - 6x + k = 0 \), let the roots be \( p \) and \( q \). For a quadratic equation \( ax^2 + bx + c = 0 \): - Sum of roots: \( p + q = -\frac{b}{a} = -\frac{-6}{1} = 6 \) - Product of roots: \( pq = \frac{c}{a} = \frac{k}{1} = k \) - Given: Difference of roots \( |p - q| = 2 \). Using the identity for the difference of roots: \[ |p - q| = \sqrt{(p + q)^2 - 4pq} \] Substitute \( p + q = 6 \) and \( pq = k \): \[ \sqrt{6^2 - 4k} = 2 \] \[ \sqrt{36 - 4k} = 2 \] Square both sides: \[ 36 - 4k = 4 \implies 4k = 36 - 4 = 32 \implies k = 8 \] Thus, the value of \( k \) is: \[ \boxed{8} \]