Solve the equation by the square root property. $$ (x-7)^{2}=36 $$

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UpStudy AI · Accepted Answer

To solve the equation $$ (x - 7)^2 = 36 $$ using the **square root property**, follow these steps:

### **Step 1: Understand the Square Root Property**
The square root property states that if $$ y^2 = a $$, then $$ y = \sqrt{a} $$ **or** $$ y = -\sqrt{a} $$.

### **Step 2: Apply the Square Root Property to the Given Equation**
Given:
$$
(x - 7)^2 = 36
$$

Take the square root of both sides:
$$
x - 7 = \pm \sqrt{36}
$$

Simplify the square root:
$$
x - 7 = \pm 6
$$

### **Step 3: Solve for $$ x $$**
This gives us two possible equations:

1. **Positive Root:**
$$
x - 7 = 6
$$
Add 7 to both sides:
$$
x = 6 + 7 = 13
$$

2. **Negative Root:**
$$
x - 7 = -6
$$
Add 7 to both sides:
$$
x = -6 + 7 = 1
$$

### **Step 4: Write the Solution Set**
The solutions to the equation are:
$$
x = 13 \quad \text{and} \quad x = 1
$$

### **Verification**
To ensure that these solutions are correct, substitute them back into the original equation.

1. **For $$ x = 13 $$:**
$$
(13 - 7)^2 = 6^2 = 36 \quad \checkmark
$$

2. **For $$ x = 1 $$:**
$$
(1 - 7)^2 = (-6)^2 = 36 \quad \checkmark
$$

Both solutions satisfy the original equation.

### **Final Answer**
$$
x = 13 \quad \text{or} \quad x = 1
$$
$$ x = 13 $$ or $$ x = 1 $$.

Solve the equation by the square root property. \( (x-7)^{2}=36 \)

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