Which equation is y = 3(x – 2)2 – (x – 5)2 rewritten in vertex form?
y = 3 (x minus seven-halves) squared minus StartFraction 27 Over 4 EndFraction
y = 2 (x minus 1) squared minus 11
y = 2 (x minus one-half) squared minus StartFraction 53 Over 4 EndFraction
y = 2 (x minus one-half) squared minus StartFraction 27 Over 2 EndFraction

We are given the equation:

\[
y = 3(x - 2)^2 - (x - 5)^2
\]

We need to simplify this equation to rewrite it in vertex form.

### Step 1: Expand each term

#### First term:
\[
3(x - 2)^2 = 3(x^2 - 4x + 4) = 3x^2 - 12x + 12
\]

#### Second term:
\[
-(x - 5)^2 = -(x^2 - 10x + 25) = -x^2 + 10x - 25
\]

### Step 2: Combine the expanded terms

\[
y = 3x^2 - 12x + 12 - x^2 + 10x - 25
\]

Now simplify by combining like terms:

\[
y = (3x^2 - x^2) + (-12x + 10x) + (12 - 25)
\]

\[
y = 2x^2 - 2x - 13
\]

### Step 3: Complete the square to rewrite in vertex form

We will complete the square on the quadratic expression \( 2x^2 - 2x - 13 \).

#### Factor out the 2 from the first two terms:

\[
y = 2(x^2 - x) - 13
\]

#### Complete the square inside the parentheses:
To complete the square, take half of the coefficient of \(x\) (which is \(-1\)), square it, and add it inside the parentheses:

\[
\left( \frac{-1}{2} \right)^2 = \frac{1}{4}
\]

So,

\[
y = 2(x^2 - x + \frac{1}{4}) - 13 - 2\left(\frac{1}{4}\right)
\]

\[
y = 2(x - \frac{1}{2})^2 - 13 - \frac{1}{2}
\]

\[
y = 2(x - \frac{1}{2})^2 - \frac{27}{2}
\]

### Final answer in vertex form:
\[
y = 2(x - \frac{1}{2})^2 - \frac{27}{2}
\]

This matches the third option:

\[
y = 2(x - \frac{1}{2})^2 - \frac{27}{2}
\]

Would you like further details on this process?

Here are five related questions you could explore:
1. How do you complete the square for a quadratic equation?
2. How does vertex form differ from standard form?
3. What is the vertex of the equation we just derived?
4. How do you convert an equation from vertex form back to standard form?
5. How does the coefficient of \(x^2\) affect the shape of a parabola?

**Tip:** Completing the square is a powerful technique not only for rewriting equations but also for solving quadratic equations and finding minimum or maximum values of parabolas.

Which equation is y = 3(x – 2)^2 – (x – 5)^2 rewritten in vertex form? y = 3 (x minus seven-halves) squared minus StartFraction 27 Over 4 EndFraction y = 2 (x minus 1) squared minus 11 y = 2 (x minus one-half) squared minus StartFraction 53 Over 4 EndFraction y = 2 (x minus one-half) squared minus StartFraction 27 Over 2 EndFraction

Explore how to rewrite the quadratic equation y = 3(x – 2)^2 – (x – 5)^2 in vertex form. This detailed solution covers key algebra concepts like completing the square, the vertex form of a parabola, and how to simplify quadratic expressions. Ideal for students in Grades 9-11.

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