The equation of the quadratic function is ![y=(x-7)(x+3)](https://tex.z-dn.net/?f=y%3D%28x-7%29%28x%2B3%29)
Explanation:
The vertex form of the quadratic function is given by
![y=a(x-h)^{2}+k](https://tex.z-dn.net/?f=y%3Da%28x-h%29%5E%7B2%7D%2Bk)
It is given that the quadratic function has a vertex at ![(2,-25)](https://tex.z-dn.net/?f=%282%2C-25%29)
The vertex is represented by the coordinate ![(h,k)](https://tex.z-dn.net/?f=%28h%2Ck%29)
Hence, substituting
in the vertex form, we get,
![y=a(x-2)^{2}-25](https://tex.z-dn.net/?f=y%3Da%28x-2%29%5E%7B2%7D-25)
Now, substituting the x - intercept
, we have,
![0=a(7-2)^{2}-25](https://tex.z-dn.net/?f=0%3Da%287-2%29%5E%7B2%7D-25)
![0=a(5)^{2}-25](https://tex.z-dn.net/?f=0%3Da%285%29%5E%7B2%7D-25)
![25=a(25)](https://tex.z-dn.net/?f=25%3Da%2825%29)
![1=a](https://tex.z-dn.net/?f=1%3Da)
Thus, the value of a is 1.
Hence, substituting
,
in the vertex form
, we get,
![y=1(x-2)^{2}-25](https://tex.z-dn.net/?f=y%3D1%28x-2%29%5E%7B2%7D-25)
![y=(x-2)^{2}-25](https://tex.z-dn.net/?f=y%3D%28x-2%29%5E%7B2%7D-25)
![y=x^2-2x+4-25](https://tex.z-dn.net/?f=y%3Dx%5E2-2x%2B4-25)
![y=x^2-2x-21](https://tex.z-dn.net/?f=y%3Dx%5E2-2x-21)
![y=(x-7)(x+3)](https://tex.z-dn.net/?f=y%3D%28x-7%29%28x%2B3%29)
Thus, the equation of the quadratic function is ![y=(x-7)(x+3)](https://tex.z-dn.net/?f=y%3D%28x-7%29%28x%2B3%29)