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The quadratic equation with the given characteristics is:
f(x) = (x + 3)*(x - 7).
We know that the vertex must be (2, -25) and it passes through (7, 0).
For a quadratic equation:
y = a*x^2 + b*x +c
The x-value of the vertex is:
x = -b/2a
Then we have:
2 = -b/2a
We also have:
-25 = a*4 + b*2 + c
And because the function passes through (7, 0) we know that 7 is one of the roots, then:
0 = a*49 + b*7 + c
Then we have 3 equations to work with:
2 = -b/2a
-25 = a*4 + b*2 + c
0 = a*49 + b*7 + c
If we subtract the third and second equations we get:
25 = a*45 + b*5
Now, with the first equation we can rewrite:
a = -b/4
Replacing that in the other equation:
25 = a*45 + b*5
25 = (-b/4)*45 + b*5 = b*(-25/4)
25*(-4/25) =b = -4
Now we know the value of b.
2 = -(-4)/2a
a = 1
Now we need to find the value of c, we have that:
0 = 1*49 + -4*7 + c
0 = 49 - 28 + c
0 = 21 + c
Then c = -21
The equation is:
[tex]y = x^2 - 4x - 21[/tex]
It can be factorized to:
f(x) = (x + 3)*(x - 7).
If you want to learn more about quadratic equations:
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