Solve sqrt{3x+1}+sqrt{5x}=sqrt{16x+1}

$$\sqrt { 3 x + 1 } + \sqrt { 5 x } = \sqrt { 16 x + 1 }$$

Solve for x

$$x=5$$

$$x=0$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{3x+1}+\sqrt{5x}\right)^{2}=\left(\sqrt{16x+1}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{3x+1}+\sqrt{5x}\right)^{2}.

$$\left(\sqrt{3x+1}\right)^{2}+2\sqrt{3x+1}\sqrt{5x}+\left(\sqrt{5x}\right)^{2}=\left(\sqrt{16x+1}\right)^{2}$$

Calculate \sqrt{3x+1} to the power of 2 and get 3x+1.

$$3x+1+2\sqrt{3x+1}\sqrt{5x}+\left(\sqrt{5x}\right)^{2}=\left(\sqrt{16x+1}\right)^{2}$$

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

$$3x+1+2\sqrt{3x+1}\sqrt{5x}+5x=\left(\sqrt{16x+1}\right)^{2}$$

Combine 3x and 5x to get 8x.

$$8x+1+2\sqrt{3x+1}\sqrt{5x}=\left(\sqrt{16x+1}\right)^{2}$$

Calculate \sqrt{16x+1} to the power of 2 and get 16x+1.

$$8x+1+2\sqrt{3x+1}\sqrt{5x}=16x+1$$

Subtract 8x+1 from both sides of the equation.

$$2\sqrt{3x+1}\sqrt{5x}=16x+1-\left(8x+1\right)$$

To find the opposite of 8x+1, find the opposite of each term.

$$2\sqrt{3x+1}\sqrt{5x}=16x+1-8x-1$$

Combine 16x and -8x to get 8x.

$$2\sqrt{3x+1}\sqrt{5x}=8x+1-1$$

Subtract 1 from 1 to get 0.

$$2\sqrt{3x+1}\sqrt{5x}=8x$$

Square both sides of the equation.

$$\left(2\sqrt{3x+1}\sqrt{5x}\right)^{2}=\left(8x\right)^{2}$$

Expand \left(2\sqrt{3x+1}\sqrt{5x}\right)^{2}.

$$2^{2}\left(\sqrt{3x+1}\right)^{2}\left(\sqrt{5x}\right)^{2}=\left(8x\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{3x+1}\right)^{2}\left(\sqrt{5x}\right)^{2}=\left(8x\right)^{2}$$

Calculate \sqrt{3x+1} to the power of 2 and get 3x+1.

$$4\left(3x+1\right)\left(\sqrt{5x}\right)^{2}=\left(8x\right)^{2}$$

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

$$4\left(3x+1\right)\times 5x=\left(8x\right)^{2}$$

Multiply 4 and 5 to get 20.

$$20\left(3x+1\right)x=\left(8x\right)^{2}$$

Use the distributive property to multiply 20 by 3x+1.

$$\left(60x+20\right)x=\left(8x\right)^{2}$$

Use the distributive property to multiply 60x+20 by x.

$$60x^{2}+20x=\left(8x\right)^{2}$$

Expand \left(8x\right)^{2}.

$$60x^{2}+20x=8^{2}x^{2}$$

Calculate 8 to the power of 2 and get 64.

$$60x^{2}+20x=64x^{2}$$

Subtract 64x^{2} from both sides.

$$60x^{2}+20x-64x^{2}=0$$

Combine 60x^{2} and -64x^{2} to get -4x^{2}.

$$-4x^{2}+20x=0$$

Factor out x.

$$x\left(-4x+20\right)=0$$

To find equation solutions, solve x=0 and -4x+20=0.

$$x=0 x=5$$

Substitute 0 for x in the equation \sqrt{3x+1}+\sqrt{5x}=\sqrt{16x+1}.

$$\sqrt{3\times 0+1}+\sqrt{5\times 0}=\sqrt{16\times 0+1}$$

Simplify. The value x=0 satisfies the equation.

$$1=1$$

Substitute 5 for x in the equation \sqrt{3x+1}+\sqrt{5x}=\sqrt{16x+1}.

$$\sqrt{3\times 5+1}+\sqrt{5\times 5}=\sqrt{16\times 5+1}$$

Simplify. The value x=5 satisfies the equation.

$$9=9$$

List all solutions of \sqrt{3x+1}+\sqrt{5x}=\sqrt{16x+1}.

$$x=0 x=5$$