As I suspected, the fact that we know the values of two possible solutions does not determine the equation.
1)
If x = 20, when P = 0, this means that 20 is one of the solutions of the equation.
Also gives the equation 400d+20e+f=0, when substituting x and P in the equation.
And that is all
INSUFF
2) If P has the greatest value when x=25, this means that x is the x of the vertex. The x of the vertex has the formula -b/2a, considering (ax^2 + bx + c)
Thus you have -e/2d = 25.
As d<0 you have e = 50d
And that is all
INSUFF
However, if you put both stmts together, you can remember that the x of vertex is exactly between the two "zeros-solutions" of the equation.
Thus, if one solution is 20, and the x of the vertex is 25, the other solution is 30.
So you might be attempt to think: Bingo! I know that when I sum the solutions I have the opposite of e, and when I multiply them I have f. But in fact, this is only valid when the term which multiply \(x^2\) is 1, or d=1.
In fact the right formula is
(considering ax^2 + bx + c; x1 and x2 as the solutions)
\(x1 + x2 = -b/a\)
\(x1*x2 = c/a\)
This means that since the solutions are 20 and 30, the equation is x^2 - 50x + 600
Or, because d<0, -x^2 + 50x -600
But it could be:
-2x^2 + 100x - 1200
-3x^2 + 150x - 1800
-4x^2 + 200x - 2400
.
.
.
When we read that f is a constant, this means that f is a number, but it can be -600, -1200, ...
f is a constant means that f, at the end of the solution will be a number and not a equation depending on other variables.
So, even with the two stmts together we only find a ration of d to e to f
ANSWER E.
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