Certainly! Let's solve the equation step-by-step.
The given equation is:
[tex]\[ x^2 + 16x = 22 \][/tex]
To convert this equation into a perfect square form, we need to complete the square. Here are the steps:
1. Identify the coefficient of the linear term (the term with [tex]\( x \)[/tex]). In this case, the coefficient is [tex]\( 16 \)[/tex].
2. Halve this coefficient, and then square the result. Halving [tex]\( 16 \)[/tex] gives [tex]\( 8 \)[/tex], and squaring [tex]\( 8 \)[/tex] gives [tex]\( 64 \)[/tex]:
[tex]\[ \left(\frac{16}{2}\right)^2 = 8^2 = 64 \][/tex]
3. Add and subtract this square (i.e., [tex]\( 64 \)[/tex]) inside the equation to maintain the equality:
[tex]\[ x^2 + 16x + 64 - 64 = 22 \][/tex]
[tex]\[ x^2 + 16x + 64 = 22 + 64 \][/tex]
4. Observe that the left-hand side is now a perfect square trinomial.
5. Rewrite the trinomial as a square of a binomial:
[tex]\[ (x + 8)^2 = 86 \][/tex]
So, the equation [tex]\( x^2 + 16x = 22 \)[/tex] is equivalent to:
[tex]\[ (x + 8)^2 = 86 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{(x+8)^2=86} \][/tex]
Thus, option B is the correct one.
