Answer:
[tex]f(x) = (x+3)\cdot (x-7)[/tex]
Step-by-step explanation:
According to the statement, the quadratic function (parabola) has the following form:
[tex]y + 25 = C \cdot (x-2)^{2}[/tex]
The standard form is unleashed after expading the algebraic equation:
[tex]y + 25 = C\cdot (x^{2}-4\cdot x +4)[/tex]
[tex]y = C\cdot x^{2} - 4\cdot C \cdot x + (4\cdot C - 25)[/tex]
The zeroes of the second-order polynomial are contained in this expression:
[tex]x = \frac{4\cdot C \pm \sqrt{16\cdot C^{2}- 4\cdot C \cdot (4\cdot C - 25)}}{2\cdot C}[/tex]
[tex]x = 2 \pm \frac{5\cdot \sqrt {C}}{ C}[/tex]
Given that [tex]x = 7[/tex] and assuming a positive sign, the value of C is finally found:
[tex]7\cdot C = 2\cdot C + 5\cdot \sqrt{C}[/tex]
[tex]5\cdot C = 5 \cdot \sqrt{C}[/tex]
[tex]C = +1[/tex] (since vertex is a minimum).
The remaining zero of the polynomial is:
[tex]x = 2 - 5[/tex]
[tex]x = -3[/tex]
Therefore, the polynomial is [tex]f(x) = (x+3)\cdot (x-7)[/tex].
