To solve the quadratic equation [tex]\[ 2(x-7)^2 - 31 = 1 \][/tex] for [tex]\( x \)[/tex] in its simplest form, follow these steps:
### Step 1: Simplify the Equation
First, move the constant term on the right-hand side to the left-hand side by subtracting 1 from both sides:
[tex]\[ 2(x-7)^2 - 31 - 1 = 0 \][/tex]
This simplifies to:
[tex]\[ 2(x-7)^2 - 32 = 0 \][/tex]
### Step 2: Isolate the Squared Term
Add 32 to both sides to isolate the squared term:
[tex]\[ 2(x-7)^2 - 32 + 32 = 32 \][/tex]
[tex]\[ 2(x-7)^2 = 32 \][/tex]
### Step 3: Solve for the Squared Expression
Divide both sides by 2 to simplify further:
[tex]\[ \frac{2(x-7)^2}{2} = \frac{32}{2} \][/tex]
[tex]\[ (x-7)^2 = 16 \][/tex]
### Step 4: Solve for [tex]\( x-7 \)[/tex]
To solve for [tex]\( x \)[/tex], take the square root of both sides. Remember that taking the square root gives us two solutions, a positive and a negative root:
[tex]\[ x-7 = \pm \sqrt{16} \][/tex]
[tex]\[ x-7 = \pm 4 \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Now solve for [tex]\( x \)[/tex] in both cases:
1. [tex]\( x-7 = 4 \)[/tex]
[tex]\[ x = 4 + 7 \][/tex]
[tex]\[ x = 11 \][/tex]
2. [tex]\( x-7 = -4 \)[/tex]
[tex]\[ x = -4 + 7 \][/tex]
[tex]\[ x = 3 \][/tex]
### Solution
The solutions to the quadratic equation [tex]\( 2(x-7)^2 - 31 = 1 \)[/tex] are:
[tex]\[ x = 11 \quad \text{and} \quad x = 3 \][/tex]
In simplest form, the solutions are:
[tex]\[ \boxed{3 \text{ and } 11} \][/tex]
