To convert the quadratic equation [tex]\( y = x^2 + 18x + 11 \)[/tex] into vertex form by completing the square, follow these steps:
1. Identify the coefficients:
[tex]\[
a = 1, \, b = 18, \, c = 11
\][/tex]
2. Write the quadratic equation in a suitable form for completing the square:
[tex]\[
y = x^2 + 18x + 11
\][/tex]
3. Complete the square for the [tex]\(x\)[/tex] terms:
- Take the coefficient of [tex]\(x\)[/tex], which is 18, divide it by 2, and then square the result:
[tex]\[
\left(\frac{18}{2}\right)^2 = 81
\][/tex]
4. Add and subtract this square inside the equation to create a perfect square trinomial:
[tex]\[
y = x^2 + 18x + 81 - 81 + 11
\][/tex]
5. Group the perfect square trinomial and combine the constants:
[tex]\[
y = (x^2 + 18x + 81) - 81 + 11
\][/tex]
6. Rewrite the perfect square trinomial as a squared binomial:
[tex]\[
y = (x + 9)^2 - 70
\][/tex]
The equation in vertex form is:
[tex]\[
y = (x + 9)^2 - 70
\][/tex]
In this form, [tex]\((x - h)^2 + k\)[/tex], the vertex is at [tex]\((h, k)\)[/tex]. Here, [tex]\(h = -9\)[/tex] and [tex]\(k = -70\)[/tex]. Be sure to include the sign of [tex]\(k\)[/tex], so in this case, [tex]\(k\)[/tex] is [tex]\(-70\)[/tex].
Therefore, the vertex form of the given equation is:
[tex]\[
y = (x + 9)^2 - 70
\][/tex]
