Solve the equation. $$ \begin{aligned} \sqrt{3 x-4} & =2 \sqrt{x-5} \ x & =[?]\end{aligned} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\sqrt{3x - 4} = 2\sqrt{x - 5}
$$

**Step 1: Determine the Domain**

Both square roots must have non-negative arguments:

1. $$ 3x - 4 \geq 0 $$  
   $$ 3x \geq 4 $$  
   $$ x \geq \frac{4}{3} $$

2. $$ x - 5 \geq 0 $$  
   $$ x \geq 5 $$

The more restrictive condition is $$ x \geq 5 $$. So, the domain is $$ x \geq 5 $$.

**Step 2: Solve the Equation**

Starting with the original equation:

$$
\sqrt{3x - 4} = 2\sqrt{x - 5}
$$

Square both sides to eliminate the square roots:

$$
(\sqrt{3x - 4})^2 = (2\sqrt{x - 5})^2
$$

$$
3x - 4 = 4(x - 5)
$$

Expand the right side:

$$
3x - 4 = 4x - 20
$$

Bring like terms together:

$$
3x - 4x = -20 + 4
$$

$$
-x = -16
$$

Multiply both sides by -1:

$$
x = 16
$$

**Step 3: Verify the Solution**

Check $$ x = 16 $$ in the original equation:

$$
\sqrt{3(16) - 4} = \sqrt{48 - 4} = \sqrt{44} = 2\sqrt{11}
$$

$$
2\sqrt{16 - 5} = 2\sqrt{11}
$$

Both sides are equal, so $$ x = 16 $$ is a valid solution.

**Final Answer:**

$$ x = 16 $$
$$ x = 16 $$

Solve the equation.