Question

question
complete the square to re-write the quadratic function in vertex form:
$y = x^2 + 9x + 1$

Explanation:

Step1: Isolate the x-terms

We start with the quadratic function \( y = x^2 + 9x + 1 \). First, we group the \( x \)-terms together: \( y=(x^{2}+9x)+1 \).

Step2: Complete the square for the x-terms

To complete the square for the expression \( x^{2}+bx \), we use the formula \( (\frac{b}{2})^{2} \). Here, \( b = 9 \), so \( (\frac{9}{2})^{2}=\frac{81}{4} \). We add and subtract this value inside the parentheses to keep the equation balanced:

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Step3: Rewrite the perfect square trinomial

The expression \( x^{2}+9x+\frac{81}{4} \) is a perfect square trinomial and can be written as \( (x + \frac{9}{2})^{2} \). Now we simplify the constant terms:

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Answer:

\( y=(x + \frac{9}{2})^{2}-\frac{77}{4} \)