Vertex Form 9Eeb40

1. **State the problem:** We need to rewrite the quadratic function $y = 3x^2 + 30x + 65$ in vertex form. 2. **Recall the vertex form:** The vertex form of a quadratic function is $$y = a(x - h)^2 + k$$ where $(h, k)$ is the vertex. 3. **Complete the square:** Start with the original function: $$y = 3x^2 + 30x + 65$$ Factor out the coefficient of $x^2$ from the first two terms: $$y = 3(x^2 + 10x) + 65$$ To complete the square inside the parentheses, take half of the coefficient of $x$, which is $10$, half is $5$, and square it: $5^2 = 25$. Add and subtract $25$ inside the parentheses: $$y = 3(x^2 + 10x + 25 - 25) + 65$$ Group the perfect square trinomial and simplify: $$y = 3((x + 5)^2 - 25) + 65$$ Distribute the $3$: $$y = 3(x + 5)^2 - 3 \times 25 + 65$$ $$y = 3(x + 5)^2 - 75 + 65$$ Simplify the constants: $$y = 3(x + 5)^2 - 10$$ 4. **Interpretation:** The vertex form is $$y = 3(x + 5)^2 - 10$$ with vertex at $(-5, -10)$. 5. **Answer:** The correct vertex form is $y = 3(x + 5)^2 - 10$. This corresponds to the option: $y = 3(x + 5)^2 - 10$.