Question

If the sum of the roots of the quadratic equation $ x^2 - 5x + k = 0 $ is 5, find the value of $ k $.

• A. 4
• B. 5
• C. 6
• D. 7

Hint:

For a quadratic equation, use the sum of roots \( -\frac{b}{a} \) and product of roots \( \frac{c}{a} \) to find unknown coefficients when given root properties.

Answer 1

The Correct Option is C

Given: The quadratic equation \[ ax^2 + bx + c = 0 \] has sum of roots \(-\frac{b}{a}\) and product of roots \(\frac{c}{a}\). For the equation: \[ x^2 - 5x + k = 0 \] we identify the coefficients as: \[ a = 1, \quad b = -5, \quad c = k \] 
Step 1: Find the sum of the roots \[ -\frac{b}{a} = -\frac{-5}{1} = 5 \] This matches the given sum of the roots. 
Step 2: Find the product of the roots \[ \frac{c}{a} = \frac{k}{1} = k \] 
Step 3: Use the product and test values of \(k\) Since the problem does not directly specify the product, we check possible values of \(k\) that yield roots with sum 5. 
Step 4: Check discriminant for real roots \[ \Delta = b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot k = 25 - 4k \] For real roots, \(\Delta \geq 0\), but this is not strictly required by the problem. 
Step 5: Test \(k = 6\) \[ x^2 - 5x + 6 = 0 \] Factoring: \[ (x - 2)(x - 3) = 0 \] Roots are \(2\) and \(3\). Check sum and product: \[ 2 + 3 = 5, \quad 2 \times 3 = 6 \] Sum matches the given value and product equals \(k\). 
Therefore, the value of \(k\) is: \[ \boxed{6} \]