An unbiased dice is thrown and the number shown is put for p in x ^ 2 + px + 2 = 0. The probability of the equation to have real roots is

We are given the quadratic equation:

\[
x^2 + px + 2 = 0
\]

where \(p\) is the result of throwing an unbiased dice. The dice can show any of the integers from 1 to 6, so \(p\) can take values in the set \(\{1, 2, 3, 4, 5, 6\}\).

For the quadratic equation to have real roots, the discriminant must be non-negative. The discriminant \(\Delta\) for a quadratic equation \(ax^2 + bx + c = 0\) is given by:

\[
\Delta = b^2 - 4ac
\]

In our case, \(a = 1\), \(b = p\), and \(c = 2\), so the discriminant becomes:

\[
\Delta = p^2 - 4(1)(2) = p^2 - 8
\]

For the equation to have real roots, we need \(\Delta \geq 0\):

\[
p^2 - 8 \geq 0
\]

\[
p^2 \geq 8
\]

\[
p \geq \sqrt{8} \approx 2.83
\]

Since \(p\) must be an integer (because it's the result of a dice throw), the possible values of \(p\) are those for which \(p \geq 3\). Thus, the possible values of \(p\) that satisfy this condition are \(\{3, 4, 5, 6\}\).

Now, let's calculate the probability. The total number of outcomes when rolling a die is 6, and the favorable outcomes (values of \(p\) that result in real roots) are 4 (\(p = 3, 4, 5, 6\)).

Therefore, the probability is:

\[
\text{Probability} = \frac{4}{6} = \frac{2}{3}
\]

### Final Answer:
The probability that the equation has real roots is \(\frac{2}{3}\).

Would you like further details or explanations?

Here are 5 related questions to expand your understanding:
1. What happens to the probability if we modify the constant term in the quadratic equation?
2. How would the problem change if we used a biased die instead of an unbiased one?
3. Can you find the exact roots for each valid value of \(p\) where the roots are real?
4. What if the equation had complex coefficients? How would the discriminant condition change?
5. How does the discriminant formula apply to higher-degree polynomials?

**Tip:** Always check the discriminant when determining the nature of the roots of a quadratic equation—it directly tells you if the roots are real or complex!

Discover how to determine the probability of real roots in the quadratic equation x^2 + px + 2 = 0, where p is a value from rolling an unbiased die. Learn about the discriminant and how it affects the nature of roots, suitable for students in Grades 9-12.

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