Find the value of K such that the following trinomials can be factored over the integers:
1. 36x^2+18x+K
2. 3x^2 - 16x+K
let's look at the discriminant.
if b^2 - 4ac is a perfect square, then it can be factored over the rationals, so we start with that.
I will do the 2nd, since it has smaller numbers
for 3x^2 - 16x + k
b^2 - 4ac
= 256-12k
= 4(64-3k)
remember we have to take the square root of that
√(4(64-3k))
= 2√(64-3k)
For 64-3k to be a perfect square
we need 3k to be 0,15,28,39,48,55,60 or 63
of those only 0,15,39,48,60, and 63 are multiples of 3
So for rationals, k could be 0, 5, 13, 16, 20 or 21
testing:
let's try k = 13
3x^2 - 16x + 13
x = (-16 ± √100)/6
= -1 or 13/3
so 3x^2 - 16x + 13 = (x+1)(3x-13)
k = 0, 5, 13, 16, 20, or 21
To find the value of K such that the given trinomials can be factored over the integers, we can use the discriminant.
1. For the trinomial 36x^2 + 18x + K, we'll work with the discriminant given by the formula: discriminant = b^2 - 4ac.
In this case, a = 36, b = 18, and c = K. Substituting these values into the formula, we get:
discriminant = (18)^2 - 4(36)(K)
discriminant = 324 - 144K
For the trinomial to be factorable over the integers, the discriminant needs to be a perfect square. So we can set the discriminant equal to some square number, say n^2:
324 - 144K = n^2
By rearranging the equation and factoring out a common factor of 4, we have:
36 - 16K = (n/2)^2
Now, we can check different values of n to find a value of K that satisfies this equation. For example, let's try n = 2:
36 - 16K = (2/2)^2
36 - 16K = 1
Solving for K:
-16K = 1 - 36
-16K = -35
K = 35/16
So, the value of K that allows the trinomial 36x^2 + 18x + K to be factored over the integers is K = 35/16.
2. Similarly, for the trinomial 3x^2 - 16x + K, we'll use the discriminant formula again:
discriminant = b^2 - 4ac
Here, a = 3, b = -16, and c = K. Plugging these values into the formula, we get:
discriminant = (-16)^2 - 4(3)(K)
discriminant = 256 - 12K
To factor the trinomial over the integers, the discriminant needs to be a perfect square. Setting the discriminant equal to some square number, suppose n^2:
256 - 12K = n^2
Rearranging the equation and factoring out a 4, we have:
64 - 3K = (n/4)^2
Again, we can try different values of n. Let's try n = 4:
64 - 3K = (4/4)^2
64 - 3K = 1
Solving for K:
-3K = 1 - 64
-3K = -63
K = -63/-3
K = 21
Therefore, the value of K that allows the trinomial 3x^2 - 16x + K to be factored over the integers is K = 21.