Let's focus on the correct interpretation of the given equation. The given equation simplifies to:
[tex]\[ (x - 4)^2 = 9 \][/tex]
This is a classic case of the square root property. Let's solve it step by step.
1. Equation to solve:
[tex]\[ (x - 4)^2 = 9 \][/tex]
2. Apply the square root property:
When you have an equation in the form [tex]\((a)^2 = b\)[/tex], you can solve for [tex]\(a\)[/tex] by taking the square root of both sides. However, remember that taking the square root leads to two solutions, one positive and one negative.
Therefore:
[tex]\[ x - 4 = \pm\sqrt{9} \][/tex]
3. Simplify the square root:
We know that [tex]\(\sqrt{9} = 3\)[/tex]. So, the equation becomes:
[tex]\[ x - 4 = \pm 3 \][/tex]
4. Consider the two cases:
Case 1:
[tex]\[ x - 4 = 3 \][/tex]
[tex]\[ x = 4 + 3 \][/tex]
[tex]\[ x = 7 \][/tex]
Case 2:
[tex]\[ x - 4 = -3 \][/tex]
[tex]\[ x = 4 - 3 \][/tex]
[tex]\[ x = 1 \][/tex]
So, the two solutions to the equation [tex]\((x - 4)^2 = 9\)[/tex] are:
[tex]\[ x = 7 \][/tex] and [tex]\[ x = 1 \][/tex]
Conclusively, the solutions for the given equation [tex]\((x - 4)^2 = 9\)[/tex] are:
[tex]\[ x = 7 \, \text{and} \, x = 1 \][/tex]
