Convert (x-5)² - 10 to its Standard Quadratic Form

To convert the quadratic function from vertex form to standard form, execute the following steps:

• Step 1: Begin with the given vertex form $f(x) = (x-5)^2 - 10$.
• Step 2: Expand $(x-5)^2$ using the formula $(a-b)^2 = a^2 - 2ab + b^2$, which results in:

$(x-5)^2 = x^2 - 2 \cdot x \cdot 5 + 5^2 = x^2 - 10x + 25$.

• Step 3: Replace the expanded form into the original function:

$f(x) = x^2 - 10x + 25 - 10$.

• Step 4: Combine like terms:

$f(x) = x^2 - 10x + 15$.

Therefore, the standard form of the function is $f(x) = x^2 - 10x + 15$.

Comparing with the given choices, the correct option is:

Choice 2: $f(x) = x^2 - 10x + 15$