Question

complete the square to re-write the quadratic function in vertex form:
y = x² + 4x + 1

Explanation:

Step1: Recall the vertex form formula

The vertex form of a quadratic function is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. To complete the square for \( y = x^2 + 4x + 1 \), we focus on the \( x^2 + 4x \) terms.

Step2: Complete the square for \( x^2 + 4x \)

To complete the square, we take half of the coefficient of \( x \), square it, and add and subtract it. The coefficient of \( x \) is 4, half of 4 is 2, and squaring 2 gives \( 2^2 = 4 \). So we rewrite the equation as:

$$
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Since \( x^2 + 4x + 4=(x + 2)^2 \) (by the perfect square formula \((a + b)^2=a^2 + 2ab + b^2\), here \( a = x \) and \( b = 2 \)), we substitute that in:

$$ y=(x + 2)^2 - 3 $$

Answer:

\( y=(x + 2)^2 - 3 \)