Question

What is the vertex form of: y = (x + 8) (x + 11)
Options:
y = x^2 + 8x + 11
y = x^2 + 19x + 88

Answer 1

To convert the given quadratic expression \(y = (x + 8)(x + 11)\) into vertex form, we first need to expand it.

1. **Expand the expression**:

\[
y = (x + 8)(x + 11) = x^2 + 11x + 8x + 88 = x^2 + 19x + 88
\]

Now, the correct expanded form of the given expression is:

\[
y = x^2 + 19x + 88
\]

2. **Options provided**:

- \(y = x^2 + 8x + 11\)
- \(y = x^2 + 19x + 88\)

The correct option based on the expansion is:

\[
y = x^2 + 19x + 88
\]

3. **Convert to vertex form**:

To express \(y = x^2 + 19x + 88\) in vertex form, we can complete the square.

- Start with the expression without the constant:

\[
y = x^2 + 19x + 88
\]

- To complete the square:
- Take half of the coefficient of \(x\) (which is 19), square it, and add/subtract it inside the equation.
- Half of 19 is \( \frac{19}{2} \), and squaring it gives \( \left(\frac{19}{2}\right)^2 = \frac{361}{4} \).

\[
y = \left(x^2 + 19x + \frac{361}{4}\right) - \frac{361}{4} + 88
\]

- Now combine the constants \(-\frac{361}{4} + 88 = 88 - \frac{361}{4} = \frac{352-361}{4} = \frac{-9}{4}\).

Putting it all together, we have:

\[
y = \left(x + \frac{19}{2}\right)^2 - \frac{9}{4}
\]

Thus, the vertex form of the quadratic is:

\[
y = \left(x + \frac{19}{2}\right)^2 - \frac{9}{4}
\]

In conclusion, among the given options, the correct expanded polynomial is:

\[
y = x^2 + 19x + 88
\]