The equation given is:

$y = x^2 + 2x + 8$

To write this equation in vertex form, we need to complete the square.

Step 1: Rewrite the equation

Rewrite the equation focusing on the $x$-terms:

$y = (x^2 + 2x) + 8$

Step 2: Complete the square

To complete the square for $x^2 + 2x$, take half of the coefficient of $x$, square it, and add and subtract it inside the parentheses.

The coefficient of $x$ is 2. Half of 2 is 1, and squaring it gives $1$.

Add and subtract 1 inside the parentheses:

$y = (x^2 + 2x + 1 - 1) + 8$

This simplifies to:

$y = ((x + 1)^2 - 1) + 8$

Step 3: Simplify the expression

Combine constants:

$y = (x + 1)^2 + 7$

Final Answer

The equation in vertex form is:

$y = (x + 1)^2 + 7$

In this form, the vertex of the parabola is $(-1, 7)$.

Would you like more details or have any questions?

Related Questions

1. How do you convert other quadratic equations to vertex form?
2. What is the significance of the vertex form of a quadratic equation?
3. How can we find the axis of symmetry from the vertex form?
4. Can this method be used to find the vertex of any parabola?
5. How does completing the square help in solving quadratic equations?

Tip

When completing the square, always remember to balance the equation by adding and subtracting the same value.