Certainly! Let's solve the equation step-by-step.
We start with the given equation:
[tex]\[ x^2 + 16x = 22 \][/tex]
We will complete the square on the left side to transform this equation into a perfect square trinomial.
1. Start by taking the coefficient of [tex]\(x\)[/tex], which is 16, and dividing it by 2. This gives:
[tex]\[ \frac{16}{2} = 8 \][/tex]
2. Next, we square the result obtained in step 1:
[tex]\[ 8^2 = 64 \][/tex]
3. Add and subtract this square (64) within the left side of the equation. This allows us to complete the square:
[tex]\[ x^2 + 16x + 64 - 64 = 22 \][/tex]
[tex]\[ x^2 + 16x + 64 = 22 + 64 \][/tex]
4. The left side of the equation is now a perfect square trinomial, which can be factored as:
[tex]\[ (x + 8)^2 = 86 \][/tex]
Thus, the equivalent equation to [tex]\( x^2 + 16x = 22 \)[/tex] is:
[tex]\[ (x + 8)^2 = 86 \][/tex]
Therefore, the correct answer is:
D. [tex]\((x + 8)^2 = 86\)[/tex]
