Answer :
Sure! Let's solve the given equation step-by-step to identify the correct resulting equation.
The given equation is:
[tex]\[ (x - 9)^2 = 81 \][/tex]
To find the solutions for [tex]\( x \)[/tex], we need to take the square root of both sides of the equation. Taking the square root of both sides, we have:
[tex]\[ \sqrt{(x - 9)^2} = \sqrt{81} \][/tex]
The square root of [tex]\( (x - 9)^2 \)[/tex] simplifies to the absolute value of [tex]\( x - 9 \)[/tex], so:
[tex]\[ |x - 9| = 9 \][/tex]
An absolute value equation [tex]\( |A| = B \)[/tex] can be split into two linear equations:
[tex]\[ A = B \quad \text{or} \quad A = -B \][/tex]
Applying this to our equation:
[tex]\[ x - 9 = 9 \quad \text{or} \quad x - 9 = -9 \][/tex]
Therefore, the equation resulting from taking the square root of both sides of [tex]\( (x - 9)^2 = 81 \)[/tex] is:
[tex]\[ x - 9 = \pm9 \][/tex]
So, the correct answer is:
[tex]\[ x - 9 = \pm 9 \][/tex]
The given equation is:
[tex]\[ (x - 9)^2 = 81 \][/tex]
To find the solutions for [tex]\( x \)[/tex], we need to take the square root of both sides of the equation. Taking the square root of both sides, we have:
[tex]\[ \sqrt{(x - 9)^2} = \sqrt{81} \][/tex]
The square root of [tex]\( (x - 9)^2 \)[/tex] simplifies to the absolute value of [tex]\( x - 9 \)[/tex], so:
[tex]\[ |x - 9| = 9 \][/tex]
An absolute value equation [tex]\( |A| = B \)[/tex] can be split into two linear equations:
[tex]\[ A = B \quad \text{or} \quad A = -B \][/tex]
Applying this to our equation:
[tex]\[ x - 9 = 9 \quad \text{or} \quad x - 9 = -9 \][/tex]
Therefore, the equation resulting from taking the square root of both sides of [tex]\( (x - 9)^2 = 81 \)[/tex] is:
[tex]\[ x - 9 = \pm9 \][/tex]
So, the correct answer is:
[tex]\[ x - 9 = \pm 9 \][/tex]
