To rewrite the given quadratic equation $y = x^2 + 8x - 2$ in vertex form, we complete the square. Here's the step-by-step process:

1. Start with the given equation: $y = x^2 + 8x - 2$

2. Focus on the quadratic and linear terms to complete the square: $y = (x^2 + 8x) - 2$

3. Find the value to complete the square: Take half the coefficient of $x$ (which is 8), square it, and add/subtract it: $\text{Half of } 8 = 4, \quad 4^2 = 16$

   Add and subtract 16 inside the parentheses: $y = (x^2 + 8x + 16 - 16) - 2$

4. Rewrite the perfect square trinomial as a square: $y = ((x + 4)^2 - 16) - 2$

5. Simplify the constants: Combine $-16$ and $-2$: $y = (x + 4)^2 - 18$

Thus, the quadratic equation in vertex form is: $y = (x + 4)^2 - 18$

Key Insights:

• The vertex of the parabola is $(-4, -18)$.
• This form makes it easy to determine transformations of the parent function $y = x^2$.

Would you like further clarification or examples of similar problems?

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Five Related Questions:

1. How do you graph a quadratic function given in vertex form?
2. What is the significance of the vertex in a quadratic equation?
3. How can you convert a quadratic equation in vertex form back to standard form?
4. How do you determine the axis of symmetry from vertex form?
5. Can the same process of completing the square be used for solving quadratic equations?

Tip:

When completing the square, always ensure the coefficient of $x^2$ is 1 before proceeding. If it's not, factor it out first!