Which equation is y = (x + 3)2 + (x + 4)2 rewritten in vertex form?
y = 2 (x + seven-halves) squared minus one-fourth
y = 2 (x + seven-halves) squared minus one-half
y = 2(x + 7)2 – 73
y = (x + 7)2 – 24

We are given the equation:

\[
y = (x + 3)^2 + (x + 4)^2
\]

First, let's expand each term on the right-hand side.

1. Expand \( (x + 3)^2 \):
   \[
   (x + 3)^2 = x^2 + 6x + 9
   \]
2. Expand \( (x + 4)^2 \):
   \[
   (x + 4)^2 = x^2 + 8x + 16
   \]

Now, add these two expanded terms together:

\[
y = (x^2 + 6x + 9) + (x^2 + 8x + 16)
\]

Simplifying:

\[
y = 2x^2 + 14x + 25
\]

Next, let's convert this equation into vertex form. Vertex form is:

\[
y = a(x - h)^2 + k
\]

We first factor out the 2 from the quadratic and linear terms:

\[
y = 2(x^2 + 7x) + 25
\]

Now, complete the square on \(x^2 + 7x\). To complete the square, take half of 7 (which is \(\frac{7}{2}\)) and square it:

\[
\left(\frac{7}{2}\right)^2 = \frac{49}{4}
\]

Add and subtract \(\frac{49}{4}\) inside the parentheses:

\[
y = 2\left(x^2 + 7x + \frac{49}{4} - \frac{49}{4}\right) + 25
\]

Simplify the expression inside the parentheses:

\[
y = 2\left(\left(x + \frac{7}{2}\right)^2 - \frac{49}{4}\right) + 25
\]

Now distribute the 2:

\[
y = 2\left(x + \frac{7}{2}\right)^2 - \frac{49}{2} + 25
\]

Convert 25 into a fraction:

\[
25 = \frac{50}{2}
\]

Finally, simplify:

\[
y = 2\left(x + \frac{7}{2}\right)^2 + \left(-\frac{49}{2} + \frac{50}{2}\right)
\]

\[
y = 2\left(x + \frac{7}{2}\right)^2 + \frac{1}{2}
\]

Thus, the equation in vertex form is:

\[
y = 2\left(x + \frac{7}{2}\right)^2 + \frac{1}{2}
\]

However, none of the answer choices match exactly. The closest match is:

**\( y = 2(x + 7/2)^2 - 1/2 \)**

Would you like further details or have any questions?

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**Related Questions:**

1. How do you complete the square on a quadratic equation?
2. What is the significance of vertex form in quadratic functions?
3. How do you find the vertex of a quadratic equation in standard form?
4. Can you explain why we factor out the 2 when completing the square?
5. How does the transformation of \(y = x^2\) change when moving to \(y = a(x - h)^2 + k\)?

**Tip:** Always double-check your arithmetic when completing the square, especially with fractions!

Which equation is y = (x + 3)^2 + (x + 4)^2 rewritten in vertex form? y = 2(x + seven-halves)^2 - 1/4 y = 2(x + seven-halves)^2 - 1/2 y = 2(x + 7)^2 - 73 y = (x + 7)^2 - 24

Learn how to rewrite the quadratic equation y = (x + 3)^2 + (x + 4)^2 in vertex form using the method of completing the square. This step-by-step solution involves algebraic techniques and provides insights into quadratic equations and vertex form. Suitable for students in Grades 9-11.

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