Solve sqrt{3x-4}=2sqrt{x-5}

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

Solve for x

$$x=16$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{3x-4} to the power of 2 and get 3x-4.

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

Expand \left(2\sqrt{x-5}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Calculate \sqrt{x-5} to the power of 2 and get x-5.

$$3x-4=4\left(x-5\right)$$

Use the distributive property to multiply 4 by x-5.

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

Subtract 4x from both sides.

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

Combine 3x and -4x to get -x.

$$-x-4=-20$$

Add 4 to both sides.

$$-x=-20+4$$

Add -20 and 4 to get -16.

$$-x=-16$$

Multiply both sides by -1.

$$x=16$$

Substitute 16 for x in the equation \sqrt{3x-4}=2\sqrt{x-5}.

$$\sqrt{3\times 16-4}=2\sqrt{16-5}$$

Simplify. The value x=16 satisfies the equation.

$$2\times 11^{\frac{1}{2}}=2\times 11^{\frac{1}{2}}$$

Equation \sqrt{3x-4}=2\sqrt{x-5} has a unique solution.

$$x=16$$