Question

What is the vertex form of: y = (x + 8) (x + 11)

Answer 1

To convert the quadratic function given in factored form \( y = (x + 8)(x + 11) \) into vertex form, we first expand it:

1. Expand the function:
\[
y = (x + 8)(x + 11) = x^2 + 11x + 8x + 88 = x^2 + 19x + 88
\]

2. Now, we need to convert \( y = x^2 + 19x + 88 \) into vertex form, which is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. We can do this by completing the square.

3. To complete the square:
- Take the coefficient of \( x \) (which is 19), divide it by 2 to get \( \frac{19}{2} \), and then square it to get \( \left(\frac{19}{2}\right)^2 = \frac{361}{4} \).
- Rewrite the quadratic as:
\[
y = x^2 + 19x + \frac{361}{4} - \frac{361}{4} + 88
\]

4. Combine the constants:
\[
88 = \frac{352}{4}
\]
So we have:
\[
y = x^2 + 19x + \frac{361}{4} - \frac{361}{4} + \frac{352}{4} = x^2 + 19x + \frac{361}{4} - \frac{9}{4}
\]
\[
y = \left(x + \frac{19}{2}\right)^2 - \frac{9}{4}
\]

Now we have the vertex form:

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

The vertex of the parabola is \(\left(-\frac{19}{2}, -\frac{9}{4}\right)\).

Answer 2

in decimal form

Answer 3

To express the vertex \(\left(-\frac{19}{2}, -\frac{9}{4}\right)\) in decimal form:

1. **Calculate \(-\frac{19}{2}\):**
\[
-\frac{19}{2} = -9.5
\]

2. **Calculate \(-\frac{9}{4}\):**
\[
-\frac{9}{4} = -2.25
\]

Thus, the vertex in decimal form is:

\[
(-9.5, -2.25)
\]

So, the vertex form of the equation \(y = (x + 8)(x + 11)\) in decimal form is:

\[
y = \left(x + 9.5\right)^2 - 2.25
\]