Solve (x^2(2x-4)^(1/2))^(1/2) (x- 2(2x-4)^(1/2))^(1/2

Question

Simplify the expression

x x 2 x - 4 - 4 x + 8

Show Solution

x_{1} = 2, x_{2} = 4

Evaluate

(x^{2} (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}} (x - 2 (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}}

To find the roots of the expression,set the expression equal to 0

(x^{2} (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}} (x - 2 (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}} = 0

Find the domain

More Steps

(x^{2} (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}} (x - 2 (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}} = 0, x \geq 2

Calculate

(x^{2} (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}} (x - 2 (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}} = 0

Multiply the terms

(x^{3} (2 x - 4)^{\frac{1}{2}} - 4 x^{3} + 8 x^{2})^{\frac{1}{2}} = 0

The only way a root could be 0 is when the radicand equals 0

x^{3} (2 x - 4)^{\frac{1}{2}} - 4 x^{3} + 8 x^{2} = 0

Factor the expression

More Steps

2 \times x^{2} (x (x - 2)^{\frac{1}{2}} - 22 \times x + 42) = 0

Elimination the left coefficient

x^{2} (x (x - 2)^{\frac{1}{2}} - 22 \times x + 42) = 0

Separate the equation into 2 possible cases

x^{2} = 0 x (x - 2)^{\frac{1}{2}} - 22 \times x + 42 = 0

The only way a power can be 0 is when the base equals 0

x = 0 x (x - 2)^{\frac{1}{2}} - 22 \times x + 42 = 0

Solve the equation

More Steps

x = 0 x = 4 x = 2

Check if the solution is in the defined range

x = 0 x = 4 x = 2, x \geq 2

Find the intersection of the solution and the defined range

x = 4 x = 2

Solution

x_{1} = 2, x_{2} = 4

Show Solution