To solve the equation [tex]\(x^2 + 2x = 19\)[/tex] by completing the square, follow these steps:
1. Identify the quadratic and linear terms: The given equation is [tex]\(x^2 + 2x = 19\)[/tex]. Notice that [tex]\(x^2\)[/tex] is the quadratic term and [tex]\(2x\)[/tex] is the linear term.
2. Add a term to both sides to complete the square: To complete the square, we need to form a perfect square trinomial on the left side. The coefficient of the linear term is 2.
3. Compute the needed term: Take half of the coefficient of [tex]\(x\)[/tex] (which is 2), divide by 2, and square it:
[tex]\[
\left(\frac{2}{2}\right)^2 = 1
\][/tex]
4. Add and subtract this term inside the equation: Add 1 to both sides of the equation to maintain equality:
[tex]\[
x^2 + 2x + 1 = 19 + 1
\][/tex]
5. Simplify both sides:
[tex]\[
x^2 + 2x + 1 = 20
\][/tex]
6. Express as a squared binomial: The left-hand side [tex]\(x^2 + 2x + 1\)[/tex] can be written as the square of a binomial:
[tex]\[
(x + 1)^2 = 20
\][/tex]
Therefore, the completed square form of the equation [tex]\(x^2 + 2x = 19\)[/tex] is [tex]\((x + 1)^2 = 20\)[/tex].
So, the correct answer is:
D. [tex]\((x + 1)^2 = 20\)[/tex]
