Question

quadratic function f can be written as $f(x)=(x - 7)^2+1$ and $f(x)=x^2+kx + 50$. what is the value of $k$?

Explanation:

Step1: Expand the vertex form

The vertex form of the quadratic function is \( f(x)=(x - 7)^2+1 \). We expand \((x - 7)^2\) using the formula \((a - b)^2=a^2-2ab + b^2\), where \(a = x\) and \(b = 7\). So \((x - 7)^2=x^2-14x + 49\). Then \(f(x)=x^2-14x + 49+1=x^2-14x + 50\).

Step2: Compare with the standard form

The other form of the quadratic function is \(f(x)=x^2+kx + 50\). By comparing the coefficients of \(x\) in \(x^2-14x + 50\) and \(x^2+kx + 50\), we can see that \(k=-14\).

Answer:

\(-14\)