consider the equation x^2+kx+1=0
a single fair die(dice)is rolled to determine the value of the middle coefficient, k. the value for k is the number of dots on the upper face of the die. what is the probability that the equation will have real, unequal roots?
for the roots to be real and distinct, the discriminant must be positive.
k^2 - 4 > 0
k > 2
so, we need to roll a 3,4,5,6
p(k>2) = 4/6 = 2/3
Ah, the dice of destiny! Let's roll with this equation and figure out the probability of having real, unequal roots.
For the equation x^2 + kx + 1 = 0, we want to find the values of k that make the roots real and unequal.
To have real roots, the discriminant (b^2 - 4ac) must be greater than zero. In this case, a=1, b=k, and c=1.
So, we have k^2 - 4 > 0. Simplifying, k^2 > 4. Taking the square root of both sides, we get |k| > 2.
Now, let's see how many possibilities we have for k on a fair die. Since there are 6 faces on a die, each with a different number of dots, we have 6 possibilities for k.
Out of these 6 possibilities, only 4 values of k will satisfy the condition |k| > 2: k = -3, -2, 2, or 3.
Thus, the probability of having real, unequal roots is 4 out of 6, or simply 2 out of 3.
And remember, even if the roots aren't real and unequal, there's always a chance for a complex and exciting solution! Keep rolling with it!
To find the probability that the equation will have real, unequal roots, we need to determine the values of k that satisfy the condition.
For a quadratic equation to have real, unequal roots (distinct roots), the discriminant, denoted as Δ, must be greater than 0.
The discriminant is given by the formula Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation (in the form ax^2 + bx + c = 0).
In this case, our equation is x^2 + kx + 1 = 0, which means a = 1, b = k, and c = 1.
Substituting these values into the discriminant formula, we get Δ = k^2 - 4.
For the equation to have real, unequal roots, Δ must be greater than 0.
So, we have the inequality k^2 - 4 > 0.
Solving this inequality, we get:
k^2 > 4.
Taking the square root of both sides (note that k is a positive integer from rolling a fair die), we have:
k > 2.
Now, we need to determine the favorable outcomes, which is the number of possible values that satisfy the inequality k > 2.
Considering a fair die, the face values are 1, 2, 3, 4, 5, and 6.
Out of these values, only 3, 4, 5, and 6 satisfy the condition k > 2.
Therefore, the favorable outcomes are 4 (k can be 3, 4, 5, or 6).
Since there are 6 equally likely outcomes (the face values of the fair die), the total number of outcomes, the probability is calculated by dividing the favorable outcomes by the total outcomes:
Probability = Favorable outcomes / Total outcomes = 4 / 6 = 2 / 3.
Hence, the probability that the equation will have real, unequal roots is 2/3.
To find the probability that the equation will have real, unequal roots, we need to determine the values of k that satisfy this condition.
The equation x^2 + kx + 1 = 0 represents a quadratic equation, and its discriminant (b^2 - 4ac) is used to determine the nature of its roots. For a quadratic equation ax^2 + bx + c = 0, if the discriminant is positive, the equation will have two real and unequal roots.
Here, the coefficient of x^2 (a) is 1, the coefficient of x (b) is k, and the constant term (c) is 1. So, the discriminant of the equation is given by:
Discriminant = (k^2) - (4)(1)(1)
= k^2 - 4
For the equation to have real, unequal roots, the discriminant must be greater than zero:
k^2 - 4 > 0
Now, we need to consider the values of k for which this inequality is true.
k^2 - 4 > 0
k^2 > 4
k > 2 or k < -2
It is given that k represents the number of dots on the upper face of a fair die. A fair die has six sides numbered from 1 to 6. So, the possible values of k are integers 1, 2, 3, 4, 5, and 6.
From the above inequality, we can see that only the values 3, 4, 5, and 6 satisfy the condition for the equation to have real, unequal roots.
Therefore, the probability of obtaining a value of k for which the equation x^2 + kx + 1 = 0 has real, unequal roots is 4 out of 6 (since there are 4 favorable outcomes out of 6 possible outcomes when rolling the die).
Thus, the probability is 4/6, which can be simplified to 2/3.