Find the value of k so that each trinomial is a perfect square

49d^2-56d+k

Hmmm. 2*sqrt(49)*sqrtk=56, right?

then sqrtk=56/(14)
and k= ????

Take the square root of 49 then multiply it by 2, which is 14

Take 56 and divide it by this number, 14 which is 4
k=4^2 which is 16

To find the value of k so that the trinomial 49d^2 - 56d + k is a perfect square, we need to determine the square root of the first term and the square root of the last term. Then, we can take the square of the middle term and check if it matches with k.

1. Square root of the first term: √(49d^2) = 7d
2. Square root of the last term: √k

Now, let's square the middle term, which is -56d:

(-56d)^2 = 3136d^2

We can see that 3136d^2 matches with 7d * -56d.

Therefore, k = 3136.

So, the value of k is 3136 to make the trinomial 49d^2 - 56d + k a perfect square.

To find the value of k that makes the trinomial a perfect square, we need to determine the perfect square trinomial that corresponds to 49d^2-56d+k.

A perfect square trinomial has the form (x ± y)^2, where x and y are constants. To match this form, let's break down the quadratic equation 49d^2-56d+k.

The first term, 49d^2, suggests that the square root of 49d^2 is 7d. To continue, we can use the fact that (a ± b)^2 = a^2 ± 2ab + b^2.

Using this pattern, we can deduce that:
49d^2 - 56d + k = (7d - 4)^2
Expanding (7d - 4)^2 gives us:
49d^2 - 56d + k = 49d^2 - 56d + 16

Since we want this expanded form to be equal to 49d^2 - 56d + k, we can conclude that k must be equal to 16.

Therefore, the value of k that makes the trinomial 49d^2-56d+k a perfect square is k = 16.