Solve 2x(sqrt{1/4-x})-3*sqrt{4-x}=0

$$2 x ( \sqrt { \frac { 1 } { 4 - x } } ) - 3 \cdot \sqrt { 4 - x } = 0$$

Solve for x

$$x = \frac{12}{5} = 2\frac{2}{5} = 2.4$$

Solution Steps

Subtract -3\sqrt{4-x} from both sides of the equation.

$$2x\sqrt{\frac{1}{4-x}}=3\sqrt{4-x}$$

Square both sides of the equation.

$$\left(2x\sqrt{\frac{1}{4-x}}\right)^{2}=\left(3\sqrt{4-x}\right)^{2}$$

Expand \left(2x\sqrt{\frac{1}{4-x}}\right)^{2}.

$$2^{2}x^{2}\left(\sqrt{\frac{1}{4-x}}\right)^{2}=\left(3\sqrt{4-x}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}\left(\sqrt{\frac{1}{4-x}}\right)^{2}=\left(3\sqrt{4-x}\right)^{2}$$

Calculate \sqrt{\frac{1}{4-x}} to the power of 2 and get \frac{1}{4-x}.

$$4x^{2}\times \frac{1}{4-x}=\left(3\sqrt{4-x}\right)^{2}$$

Express 4\times \frac{1}{4-x} as a single fraction.

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

Expand \left(3\sqrt{4-x}\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

$$\frac{4}{4-x}x^{2}=9\left(\sqrt{4-x}\right)^{2}$$

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

$$\frac{4}{4-x}x^{2}=9\left(4-x\right)$$

Use the distributive property to multiply 9 by 4-x.

$$\frac{4}{4-x}x^{2}=36-9x$$

Express \frac{4}{4-x}x^{2} as a single fraction.

$$\frac{4x^{2}}{4-x}=36-9x$$

Multiply both sides of the equation by -x+4.

$$4x^{2}=\left(-x+4\right)\times 36-9x\left(-x+4\right)$$

Use the distributive property to multiply -x+4 by 36.

$$4x^{2}=-36x+144-9x\left(-x+4\right)$$

Use the distributive property to multiply -9x by -x+4.

$$4x^{2}=-36x+144+9x^{2}-36x$$

Combine -36x and -36x to get -72x.

$$4x^{2}=-72x+144+9x^{2}$$

Add 72x to both sides.

$$4x^{2}+72x=144+9x^{2}$$

Subtract 144 from both sides.

$$4x^{2}+72x-144=9x^{2}$$

Subtract 9x^{2} from both sides.

$$4x^{2}+72x-144-9x^{2}=0$$

Combine 4x^{2} and -9x^{2} to get -5x^{2}.

$$-5x^{2}+72x-144=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -5x^{2}+ax+bx-144. To find a and b, set up a system to be solved.

$$a+b=72 ab=-5\left(-144\right)=720$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 720.

$$1,720 2,360 3,240 4,180 5,144 6,120 8,90 9,80 10,72 12,60 15,48 16,45 18,40 20,36 24,30$$

Calculate the sum for each pair.

$$1+720=721 2+360=362 3+240=243 4+180=184 5+144=149 6+120=126 8+90=98 9+80=89 10+72=82 12+60=72 15+48=63 16+45=61 18+40=58 20+36=56 24+30=54$$

The solution is the pair that gives sum 72.

$$a=60 b=12$$

Rewrite -5x^{2}+72x-144 as \left(-5x^{2}+60x\right)+\left(12x-144\right).

$$\left(-5x^{2}+60x\right)+\left(12x-144\right)$$

Factor out 5x in the first and -12 in the second group.

$$5x\left(-x+12\right)-12\left(-x+12\right)$$

Factor out common term -x+12 by using distributive property.

$$\left(-x+12\right)\left(5x-12\right)$$

To find equation solutions, solve -x+12=0 and 5x-12=0.

$$x=12 x=\frac{12}{5}$$

Substitute 12 for x in the equation 2x\sqrt{\frac{1}{4-x}}-3\sqrt{4-x}=0. The expression \sqrt{\frac{1}{4-12}} is undefined because the radicand cannot be negative.

$$2\times 12\sqrt{\frac{1}{4-12}}-3\sqrt{4-12}=0$$

Substitute \frac{12}{5} for x in the equation 2x\sqrt{\frac{1}{4-x}}-3\sqrt{4-x}=0.

$$2\times \frac{12}{5}\sqrt{\frac{1}{4-\frac{12}{5}}}-3\sqrt{4-\frac{12}{5}}=0$$

Simplify. The value x=\frac{12}{5} satisfies the equation.

$$0=0$$

Equation 2\sqrt{\frac{1}{4-x}}x=3\sqrt{4-x} has a unique solution.

$$x=\frac{12}{5}$$