From beginner to expert, IDNLearn.com has answers for everyone. Find reliable solutions to your questions quickly and accurately with help from our dedicated community of experts.
Sagot :
Let's solve the problem step-by-step to determine which equation is equivalent to the given equation: [tex]\(x^2 + 16x = 22\)[/tex].
First, we'll rewrite the given equation in the form of a perfect square.
### Step 1: Add and subtract the same value to form a perfect square
Consider the given equation:
[tex]\[ x^2 + 16x = 22 \][/tex]
To form a perfect square, we need to add and subtract [tex]\(64\)[/tex], because [tex]\(\left(\frac{16}{2}\right)^2 = 64\)[/tex].
### Step 2: Rewrite the equation
[tex]\[ x^2 + 16x + 64 - 64 = 22 \][/tex]
[tex]\[ (x + 8)^2 - 64 = 22 \][/tex]
### Step 3: Move the constant term to the right side
[tex]\[ (x + 8)^2 - 64 = 22 \][/tex]
[tex]\[ (x + 8)^2 = 22 + 64 \][/tex]
[tex]\[ (x + 8)^2 = 86 \][/tex]
Given this transformation, we can see that the equation [tex]\( (x + 8)^2 = 86 \)[/tex] is equivalent to the given equation [tex]\( x^2 + 16x = 22 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{(x+8)^2=86} \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{C} \][/tex]
First, we'll rewrite the given equation in the form of a perfect square.
### Step 1: Add and subtract the same value to form a perfect square
Consider the given equation:
[tex]\[ x^2 + 16x = 22 \][/tex]
To form a perfect square, we need to add and subtract [tex]\(64\)[/tex], because [tex]\(\left(\frac{16}{2}\right)^2 = 64\)[/tex].
### Step 2: Rewrite the equation
[tex]\[ x^2 + 16x + 64 - 64 = 22 \][/tex]
[tex]\[ (x + 8)^2 - 64 = 22 \][/tex]
### Step 3: Move the constant term to the right side
[tex]\[ (x + 8)^2 - 64 = 22 \][/tex]
[tex]\[ (x + 8)^2 = 22 + 64 \][/tex]
[tex]\[ (x + 8)^2 = 86 \][/tex]
Given this transformation, we can see that the equation [tex]\( (x + 8)^2 = 86 \)[/tex] is equivalent to the given equation [tex]\( x^2 + 16x = 22 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{(x+8)^2=86} \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{C} \][/tex]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.