To solve for the value of k in the polynomial x² + kx + 6x² + 2k + 1, we first need to simplify the polynomial. Combining like terms, we get:
7x² + kx + 2k + 1 = 0
The sum of the roots (zeroes) of a quadratic equation ax² + bx + c = 0 is given by -b/a, and the product of the roots is given by c/a. Here, a = 7, b = k, and c = 2k + 1.
Thus, the sum of the roots is:
Sum = -k/7
And the product of the roots is:
Product = (2k + 1)/7
According to the problem, the sum of the roots should be equal to half of their product. This gives us the equation:
-k/7 = 1/2 * (2k + 1)/7
Multiplying both sides by 7 to eliminate the denominator results in:
-k = 1/2 * (2k + 1)
Multiplying through by 2 to clear the fraction:
-2k = 2k + 1
Rearranging gives:
-2k – 2k = 1
-4k = 1
So, dividing both sides by -4 leads us to:
k = -1/4
In conclusion, the value of k such that the sum of the zeroes of the polynomial equals half of their product is -1/4.