Question

Consider the quadratic equation \[ x^2+ax+b=0, \] where \(a,b\in\{1,2,3,4\}\). What is the probability that the equation has real roots?

• A. \(\dfrac{9}{16}\)
• B. \(\dfrac{5}{16}\)
• C. \(\dfrac{7}{16}\)
• D. \(\dfrac{3}{8}\)

Hint:

For a quadratic equation \[ ax^2+bx+c=0, \] the roots are real if \[ \boxed{b^2-4ac\ge0.} \] Here, \[ x^2+ax+b=0 \] so the discriminant is \[ \boxed{a^2-4b.} \]

Answer 1

The Correct Option is C

Step 1: Find the total number of cases. Since \[ a,b\in\{1,2,3,4\}, \] there are \[ 4\times4=16 \] possible ordered pairs.

Step 2: Use the condition for real roots. A quadratic equation has real roots if \[ a^2-4b\ge0. \] Check each value of \(a\): \[ \begin{array}{|c|c|c|} \hline a & a^2 & \text{Allowed values of }b \hline 1 & 1 & \text{None} 2 & 4 & 1 3 & 9 & 1,2 4 & 16 & 1,2,3,4 \hline \end{array} \] Hence, the favourable pairs are \[ 1+2+4=7. \]

Step 3: Calculate the probability. \[ P = \frac{7}{16}. \]

Step 4: Final conclusion. \[ \boxed{\frac{7}{16}} \] Hence, the correct option is \(\boxed{(C)}\).