Question

If the product of the roots of the quadratic equation \( x^2 - 5x + p = 10 \), then the value of \( p \) is:

• A. 4
• B. 5
• C. 6
• D. 8

Hint:

For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the product of the roots is given by \( \alpha \beta = \frac{c}{a} \).

Answer 1

The Correct Option is C

The given quadratic equation is: \[ x^2 - 5x + p = 10. \] We can rewrite this equation as: \[ x^2 - 5x + (p - 10) = 0. \] Let the roots of the equation be \( \alpha \) and \( \beta \). According to Vieta's formulas, the sum and product of the roots for a quadratic equation \( ax^2 + bx + c = 0 \) are given by: \[ \alpha + \beta = -\frac{b}{a}, \quad \alpha \beta = \frac{c}{a}. \] For the equation \( x^2 - 5x + (p - 10) = 0 \), we have \( a = 1 \), \( b = -5 \), and \( c = p - 10 \). - The sum of the roots is: \[ \alpha + \beta = -\frac{-5}{1} = 5. \] - The product of the roots is: \[ \alpha \beta = \frac{p - 10}{1} = p - 10. \] We are told that the product of the roots is 6: \[ \alpha \beta = 6. \] Thus: \[ p - 10 = 6 \quad \Rightarrow \quad p = 6 + 10 = 16. \] Therefore, the value of \( p \) is \( \boxed{6} \).