Questions: Solve: 3 √(x-2) + 1 = 13 Simplify the expression completely (-2 ± √(-40))/2

Transcript text: 10 Solve: $3 \sqrt{x-2}+1=13$ 11 Simplify the expression completely $\frac{-2 \pm \sqrt{-40}}{2}$.

Solution

Solution Steps

Step 1: Isolate the Square Root

Start with the equation: \[ 3 \sqrt{x-2} + 1 = 13 \] Subtract 1 from both sides: \[ 3 \sqrt{x-2} = 12 \] Then, divide both sides by 3: \[ \sqrt{x-2} = 4 \]

Step 2: Square Both Sides

Square both sides to eliminate the square root: \[ x - 2 = 16 \]

Step 3: Solve for $x$

Add 2 to both sides to find $x$: \[ x = 18 \]

Step 4: Simplify the Expression

Consider the expression: \[ \frac{-2 \pm \sqrt{-40}}{2} \] First, simplify $\sqrt{-40}$: \[ \sqrt{-40} = \sqrt{40} \cdot i = \sqrt{4 \cdot 10} \cdot i = 2\sqrt{10} \cdot i \] Now substitute back into the expression: \[ \frac{-2 \pm 2\sqrt{10}i}{2} \] This simplifies to: \[ -1 \pm \sqrt{10}i \]

Step 5: Final Result

The simplified expression yields two results: \[ -1 + \sqrt{10}i \quad \text{and} \quad -1 - \sqrt{10}i \]