Question

complete the square to re - write the quadratic function in vertex form: $y = x^{2}+7x + 6$

Explanation:

Step1: Recall vertex form and complete the square

The vertex form of a quadratic function is \(y = a(x - h)^2 + k\). For the given function \(y = x^2 + 7x + 6\), we take the coefficient of \(x\), which is \(7\). Half of \(7\) is \(\frac{7}{2}\), and squaring it gives \((\frac{7}{2})^2=\frac{49}{4}\). We add and subtract this value inside the equation:
\[

$$\begin{align*} y&=x^2 + 7x+\frac{49}{4}-\frac{49}{4}+ 6\\ &=(x^2 + 7x+\frac{49}{4})-\frac{49}{4}+ 6 \end{align*}$$

\]

Step2: Rewrite the perfect square trinomial and simplify constants

The expression \(x^2 + 7x+\frac{49}{4}\) is a perfect square trinomial and can be written as \((x + \frac{7}{2})^2\). Now we simplify the constant terms: \(-\frac{49}{4}+ 6=-\frac{49}{4}+\frac{24}{4}=-\frac{25}{4}\). So the equation becomes:
\[y=(x + \frac{7}{2})^2-\frac{25}{4}\]

Answer:

\(y=(x + \frac{7}{2})^2-\frac{25}{4}\)