Question

Complete the square to re-write the quadratic function in vertex form:
y, equals, x, squared, plus, 10, x, plus, 1
y=x
2
+10x+1

Answer 1

To write the quadratic function \( y = x^2 + 10x + 1 \) in vertex form by completing the square, follow these steps:

1. **Identify the quadratic and linear terms**: Here, the terms are \( x^2 + 10x \).

2. **Take half of the coefficient of \( x \)**: The coefficient of \( x \) is 10. Half of it is \( \frac{10}{2} = 5 \).

3. **Square that result**: \( 5^2 = 25 \).

4. **Add and subtract this square inside the equation**:
\[
y = (x^2 + 10x + 25) - 25 + 1
\]

5. **Rewrite the equation**: The expression \( (x^2 + 10x + 25) \) can be factored as \( (x + 5)^2 \).
\[
y = (x + 5)^2 - 25 + 1
\]
\[
y = (x + 5)^2 - 24
\]

So, the vertex form of the quadratic function is:
\[
y = (x + 5)^2 - 24
\]

The vertex of the parabola represented by this function is at the point \( (-5, -24) \).