find the value of k that will make x²+16x+k a perfect square
In order for x² + 16x + k to be a perfect square, we need to complete the square.
We know that (x + h)² = x² + 2hx + h².
So, to make x² + 16x + k a perfect square, we need to find a value of k such that 16x is equal to 2hx.
This means that 16 = 2h.
Simplifying, we find that h = 8.
Therefore, k = h² = 8² = 64.
So, the value of k that will make x² + 16x + k a perfect square is 64.
To find the value of k that will make x² + 16x + k a perfect square, we can complete the square.
Step 1: Identify the coefficient of x and divide it by 2.
The coefficient of x in this case is 16, so we divide it by 2 to get 8.
Step 2: Square the result from Step 1.
8 squared is 64.
Step 3: Add the result from Step 2 to both sides of the equation.
We add 64 to both sides of the equation x² + 16x + k + 64 = x² + 16x + 64.
Step 4: Simplify the equation.
The left side of the equation becomes a perfect square trinomial: (x + 8)².
So, the equation becomes (x + 8)² = x² + 16x + 64.
Step 5: From the equation, we can see that k = 64.
Therefore, the value of k that will make x² + 16x + k a perfect square is 64.
To find the value of k that will make x² + 16x + k a perfect square, we can use a well-known formula for completing the square.
The formula states that for any quadratic of the form x² + bx + c, the perfect square form can be written as (x + (b/2))² + (c - (b/2)²).
In our case, we have x² + 16x + k. So, we want to express this quadratic in the form of (x + (b/2))² + (c - (b/2)²).
Comparing the given quadratic x² + 16x + k with the perfect square form (x + (b/2))² + (c - (b/2)²), we can see that b = 16 and c = k.
Using the formula, we can substitute these values into the expression:
(x + (16/2))² + (k - (16/2)²)
Simplifying, we get:
(x + 8)² + (k - 64)
Now, we want this expression to be a perfect square, which means that the second term (k - 64) should be equal to zero. This is because any number minus itself (in this case, 64) will be zero, resulting in a perfect square.
So, we can set (k - 64) = 0 and solve for k:
k - 64 = 0
k = 64
Therefore, the value of k that will make x² + 16x + k a perfect square is k = 64.