SOLUTION: Let f(x)=x^2+ax+b for some value of a and b. f(x_0) = 0, f(x_1) = 0 and x_0x_1 = 3, x_0 +x_1 = −2. Find the minimum value of the function.
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Question 1152532: Let f(x)=x^2+ax+b for some value of a and b. f(x_0) = 0, f(x_1) = 0 and x_0x_1 = 3, x_0 +x_1 = −2. Find the minimum value of the function.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
I'm going to replace 'a' and 'b' in with 'p' and 'q' respectively.
So we have
turn into
The reason for these substitutions is to avoid confusion with the 'a' and 'b' in which is a common general form of quadratics used in many math textbooks.
If we compare and , we see that
a = 1
b = p
c = q
Use the quadratic case of Vieta's formulas
Rule: If and are roots to , then and
More info:
https://en.wikipedia.org/wiki/Vieta%27s_formulas
In this case, and play the role of and (since plugging either x value into f(x) yields f(x) = 0)
Plug in the given roots
Plug in b = p and a = 1.
The sum of the roots is equal to -2 (given).
--------------------------------------------
Move to the other part of Vieta's formula
Plug in the given roots
Plug in c = q and a = 1.
The product of the roots is 3 (given)
--------------------------------------------
With p = 2 and q = 3, we can then say,
The equation is in the form with
a = 1
b = 2
c = 3
Find the x coordinate of the vertex
The x coordinate of the vertex is -1. Plug this into the f(x) function to find the y coordinate of the vertex.
The vertex is (h,k) = (-1,2)
Graph of
We see the lowest point is (-1,2) which was the vertex we found earlier.
The final answer is 2 as this is the lowest or smallest output possible of the f(x) function. That's what it means when the teacher wants the minimum value of the function.
Recall that y = f(x) is the output, while x is the input.
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