Answer:
-21
Explanation:
You want the minimum of f(x) = x² -8x -5.
Vertex
The minimum of the function is found at the vertex of its graph. That vertex lies on the line of symmetry. There are several ways you can find its coordinates.
Vertex form
The vertex form of the function's equation is ...
f(x) = (x -h)² +k . . . . . . vertex at (h, k); k is the minimum value of f(x)
Expanding this form gives ...
f(x) = x² -2hx +h² +k
The value of h will be half the opposite of the x-coefficient:
h = -1/2(-8) = 4
This tells us ...
h² +k = -5 = 4² +k
k = -5 -16 = -21
The minimum value is -21.
Line of symmetry
The line of symmetry of the quadratic ax²+bx+c is x = -b/(2a). Here, that means ...
x = -(-8)/(2·1) = 4
Then the minimum value of the function is ...
f(4) = 4² -8·4 -5 = (4 -8)·4 -5 = -16 -5
f(4) = -21 . . . the minimum value
Graph
We find the easiest way to determine the minimum is to graph the function using a graphing calculator. The result is shown in the attachment. The minimum is -21.
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Additional comment
With a little bit of practice, you can find h²+k = c ⇒ k = c -(b/2)² using mental arithmetic.
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