Find all values of k so each trinomial can be factored using integers.
3s^2 +ks - 14
3S + KS - 14.
A*C=3*-14 =-42 = -3*14 = -1*42 = 1*-42 = -2*21= 2*-21 = -6*7 = 6*-7.
K = 3 - 14 = -11.
Solve the Quad Eq and get:
S = 4 2/3, S = -1.
(S-4 2/3)(S+1) = 0,
(S-14/3)(S+1) = 0,
Multiply both terms inside the 1st pa-
renthesis by 3:
(3S-14)(S+1).
K = -3 + 14 = 11.
(S-1)(S+4 2/3) = 0,
(S-1)(3S+14) = 0.
K = -1 + 42 = 41
(S-1/3)(S+14) = 0,
(3S-1)(S+14) = 0.
K = 1 - 42 = -41.
(S-14)(S+1/3) = 0,
(S-14)(3S+1) = 0.
K = -2 + 21 = 19.
(X-2/3)(X+7) = 0,
(3X-2)(X+7) = 0.
K = 2-21 = -19.
(X-7)(X+2/3) = 0,
(X-7)(3X+2) = 0.
K = -6 + 7 = 1.
(X-2)(X+7/3) = 0,
(X-2)(3X+7) = 0.
K = 6 - 7 = -1.
(X-7/3)(X+2) = 0,
3X-7)(X+2) = 0.
Therefore,there are 8 values of k:
k = +-11, +-41, +-19, +-1.
qdwefff
To find all values of k such that the trinomial 3s^2 + ks - 14 can be factored using integers, we need to determine if the trinomial can be factored using the factors of 3 and -14.
First, we look at the coefficient of the s^2 term, which is 3. The factors of 3 are 1 and 3.
Next, we look at the constant term, which is -14. The factors of -14 that we need to consider are -1, 1, -2, 2, -7, and 7.
To determine if the trinomial can be factored using the integers k corresponding to these factors, we find the combinations of k and check if their sum or difference is equal to the coefficient of the s term, which is k.
Let's start with the factors of 3.
If we take k = 1, the sum or difference of the factors of -14 will not give us 1.
If we take k = 3, the sum or difference of the factors of -14 will not give us 3.
Next, let's try the factors of -14.
If we take k = -1, the sum or difference of the factors of -14 is -1 + 2 = 1, which is equal to k. So, k = -1 is a valid solution.
If we take k = 1, the sum or difference of the factors of -14 is -1 - 2 = -3, which is not equal to k = 1.
If we take k = -2, the sum or difference of the factors of -14 will not give us -2.
If we take k = 2, the sum or difference of the factors of -14 is -7 + 2 = -5, which is not equal to k = 2.
If we take k = -7, the sum or difference of the factors of -14 is -1 - 7 = -8, which is not equal to k = -7.
If we take k = 7, the sum or difference of the factors of -14 is -7 - 2 = -9, which is not equal to k = 7.
Therefore, the only value of k for which the trinomial 3s^2 + ks - 14 can be factored using integers is k = -1.