Answer: Let's assume that the zeros of the given quadratic equation are a and 1/a, since they are reciprocals of each other. By the sum-product relationships for zeros of a quadratic equation, we have:
a + 1/a = -b/a1 = -4/(1-3k) (where b = 4 and a1 = 1-3k)
Multiplying both sides by a1, we get:
a1(a + 1/a) = -4
Expanding the left-hand side, we get:
a1a + a1/a = -4
Multiplying both sides by a, we get:
a1a² + a1 = -4a
Substituting a1 = 1-3k, we get:
(1-3k)a² + (1-3k) = -4a
Rearranging, we get:
(1-3k)a² + 4a + (3k-1) = 0
Since a is one of the zeros of the quadratic, we know that the quadratic can be factored as:
(1-3k)(a - r)(a - s) = 0
where r and s are the other two roots of the quadratic. By expanding the left-hand side, we get:
(1-3k)(a² - (r+s)a + rs) = 0
Comparing with the expanded form of the quadratic, we see that:
r + s = -4/(1-3k)
rs = (3k-1)/(1-3k)
By Vieta's formulas, we know that rs = c/a1, where c is the constant term of the quadratic. Substituting c = -5 and a1 = 1-3k, we get:
rs = -5/(1-3k)
Equating this with the expression we obtained earlier for rs, we get:
-5/(1-3k) = (3k-1)/(1-3k)
Multiplying both sides by 1-3k and simplifying, we get:
-5 = (3k-1)(3k-2)
Expanding the right-hand side, we get:
-5 = 9k² - 15k + 2
Simplifying, we get:
9k² - 15k - 7 = 0
Using the quadratic formula, we get:
k = (15 ± sqrt(15² + 497))/18
k = (15 ± sqrt(429))/18
Therefore, the two possible values of k are:
k = (15 + sqrt(429))/18 ≈ 1.57
k = (15 - sqrt(429))/18 ≈ -0.24
Note that both of these values of k satisfy the condition that the zeros of the quadratic are reciprocals of each other.
Explanation: