Solve by using the square root property: $$ (x+8)^{2}=81 $$

Question

UpStudy AI · Accepted Answer

To solve the equation $$(x + 8)^2 = 81$$ using the **square root property**, follow these steps:

### Step 1: Apply the Square Root Property
The square root property states that if $$ (A)^2 = B $$, then $$ A = \sqrt{B} $$ or $$ A = -\sqrt{B} $$.

Applying this to the equation:
$$
(x + 8)^2 = 81
$$
$$
x + 8 = \sqrt{81} \quad \text{or} \quad x + 8 = -\sqrt{81}
$$

### Step 2: Simplify the Square Root
$$
\sqrt{81} = 9
$$
So, the equation becomes:
$$
x + 8 = 9 \quad \text{or} \quad x + 8 = -9
$$

### Step 3: Solve for $$ x $$ in Each Case

1. **First Case:**
   $$
   x + 8 = 9
   $$
   Subtract 8 from both sides:
   $$
   x = 9 - 8
   $$
   $$
   x = 1
   $$

2. **Second Case:**
   $$
   x + 8 = -9
   $$
   Subtract 8 from both sides:
   $$
   x = -9 - 8
   $$
   $$
   x = -17
   $$

### Step 4: Write the Solution
The solutions to the equation $$(x + 8)^2 = 81$$ are:
$$
x = 1 \quad \text{and} \quad x = -17
$$

**Verification:**

1. **For $$ x = 1 $$:**
   $$
   (1 + 8)^2 = (9)^2 = 81
   $$
   
2. **For $$ x = -17 $$:**
   $$
   (-17 + 8)^2 = (-9)^2 = 81
   $$

Both solutions satisfy the original equation.
The solutions are $$ x = 1 $$ and $$ x = -17 $$.

Solve by using the square root property: \( (x+8)^{2}=81 \)