Solve sqrt{x-4}-sqrt{4x-27}+sqrt{x-9}=0

$$\sqrt{ x-4 } - \sqrt{ 4x-27 } + \sqrt{ x-9 } =0$$

Solve for x

$$x=13$$

Solution Steps

Subtract -\sqrt{4x-27}+\sqrt{x-9} from both sides of the equation.

$$\sqrt{x-4}=-\left(-\sqrt{4x-27}+\sqrt{x-9}\right)$$

To find the opposite of -\sqrt{4x-27}+\sqrt{x-9}, find the opposite of each term.

$$\sqrt{x-4}=-\left(-\sqrt{4x-27}\right)-\sqrt{x-9}$$

The opposite of -\sqrt{4x-27} is \sqrt{4x-27}.

$$\sqrt{x-4}=\sqrt{4x-27}-\sqrt{x-9}$$

Square both sides of the equation.

$$\left(\sqrt{x-4}\right)^{2}=\left(\sqrt{4x-27}-\sqrt{x-9}\right)^{2}$$

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

$$x-4=\left(\sqrt{4x-27}-\sqrt{x-9}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{4x-27}-\sqrt{x-9}\right)^{2}.

$$x-4=\left(\sqrt{4x-27}\right)^{2}-2\sqrt{4x-27}\sqrt{x-9}+\left(\sqrt{x-9}\right)^{2}$$

Calculate \sqrt{4x-27} to the power of 2 and get 4x-27.

$$x-4=4x-27-2\sqrt{4x-27}\sqrt{x-9}+\left(\sqrt{x-9}\right)^{2}$$

Calculate \sqrt{x-9} to the power of 2 and get x-9.

$$x-4=4x-27-2\sqrt{4x-27}\sqrt{x-9}+x-9$$

Combine 4x and x to get 5x.

$$x-4=5x-27-2\sqrt{4x-27}\sqrt{x-9}-9$$

Subtract 9 from -27 to get -36.

$$x-4=5x-36-2\sqrt{4x-27}\sqrt{x-9}$$

Subtract 5x-36 from both sides of the equation.

$$x-4-\left(5x-36\right)=-2\sqrt{4x-27}\sqrt{x-9}$$

To find the opposite of 5x-36, find the opposite of each term.

$$x-4-5x+36=-2\sqrt{4x-27}\sqrt{x-9}$$

Combine x and -5x to get -4x.

$$-4x-4+36=-2\sqrt{4x-27}\sqrt{x-9}$$

Add -4 and 36 to get 32.

$$-4x+32=-2\sqrt{4x-27}\sqrt{x-9}$$

Square both sides of the equation.

$$\left(-4x+32\right)^{2}=\left(-2\sqrt{4x-27}\sqrt{x-9}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-4x+32\right)^{2}.

$$16x^{2}-256x+1024=\left(-2\sqrt{4x-27}\sqrt{x-9}\right)^{2}$$

Expand \left(-2\sqrt{4x-27}\sqrt{x-9}\right)^{2}.

$$16x^{2}-256x+1024=\left(-2\right)^{2}\left(\sqrt{4x-27}\right)^{2}\left(\sqrt{x-9}\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$16x^{2}-256x+1024=4\left(\sqrt{4x-27}\right)^{2}\left(\sqrt{x-9}\right)^{2}$$

Calculate \sqrt{4x-27} to the power of 2 and get 4x-27.

$$16x^{2}-256x+1024=4\left(4x-27\right)\left(\sqrt{x-9}\right)^{2}$$

Calculate \sqrt{x-9} to the power of 2 and get x-9.

$$16x^{2}-256x+1024=4\left(4x-27\right)\left(x-9\right)$$

Use the distributive property to multiply 4 by 4x-27.

$$16x^{2}-256x+1024=\left(16x-108\right)\left(x-9\right)$$

Apply the distributive property by multiplying each term of 16x-108 by each term of x-9.

$$16x^{2}-256x+1024=16x^{2}-144x-108x+972$$

Combine -144x and -108x to get -252x.

$$16x^{2}-256x+1024=16x^{2}-252x+972$$

Subtract 16x^{2} from both sides.

$$16x^{2}-256x+1024-16x^{2}=-252x+972$$

Combine 16x^{2} and -16x^{2} to get 0.

$$-256x+1024=-252x+972$$

Add 252x to both sides.

$$-256x+1024+252x=972$$

Combine -256x and 252x to get -4x.

$$-4x+1024=972$$

Subtract 1024 from both sides.

$$-4x=972-1024$$

Subtract 1024 from 972 to get -52.

$$-4x=-52$$

Divide both sides by -4.

$$x=\frac{-52}{-4}$$

Divide -52 by -4 to get 13.

$$x=13$$

Substitute 13 for x in the equation \sqrt{x-4}-\sqrt{4x-27}+\sqrt{x-9}=0.

$$\sqrt{13-4}-\sqrt{4\times 13-27}+\sqrt{13-9}=0$$

Simplify. The value x=13 satisfies the equation.

$$0=0$$

Equation \sqrt{x-4}=\sqrt{4x-27}-\sqrt{x-9} has a unique solution.

$$x=13$$