To find the value of k, we can use the fact that the sum of the roots of a quadratic equation is equal to the negation of the coefficient of the linear term (5x) divided by the coefficient of the quadratic term (2x^2).
In this case, the sum of the roots alpha and beta is equal to -5/2.
Now, we can use the fact that the sum of the squares of the roots is equal to the square of the sum of the roots minus twice their product.
So, (alpha^2 + beta^2) = (-5/2)^2 - 2(alpha * beta).
We also know that (alpha * beta) is equal to the constant term (k) divided by the coefficient of the quadratic term (2).
Therefore, we can rewrite the equation as:
(alpha^2 + beta^2) = (-5/2)^2 - 2(k/2).
Simplifying further:
(alpha^2 + beta^2) = 25/4 - k.
Since we are given that (alpha^2 + beta^2 + alpha * beta) = 0, we can substitute this into the equation:
Answers & Comments
Answer:
To find the value of k, we can use the fact that the sum of the roots of a quadratic equation is equal to the negation of the coefficient of the linear term (5x) divided by the coefficient of the quadratic term (2x^2).
In this case, the sum of the roots alpha and beta is equal to -5/2.
Now, we can use the fact that the sum of the squares of the roots is equal to the square of the sum of the roots minus twice their product.
So, (alpha^2 + beta^2) = (-5/2)^2 - 2(alpha * beta).
We also know that (alpha * beta) is equal to the constant term (k) divided by the coefficient of the quadratic term (2).
Therefore, we can rewrite the equation as:
(alpha^2 + beta^2) = (-5/2)^2 - 2(k/2).
Simplifying further:
(alpha^2 + beta^2) = 25/4 - k.
Since we are given that (alpha^2 + beta^2 + alpha * beta) = 0, we can substitute this into the equation:
0 = 25/4 - k.
Solving for k:
k = 25/4.
Therefore, the value of k is 25/4.
HOPE ITS HELP YOU
PLEASE THANK ME