Solve sqrt{5x+1}-sqrt{2x+1}=2

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

Solve for x

$$x = \frac{56}{9} = 6\frac{2}{9} \approx 6.222222222$$

Solution Steps

Subtract -\sqrt{2x+1} from both sides of the equation.

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

Square both sides of the equation.

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

Calculate \sqrt{5x+1} to the power of 2 and get 5x+1.

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

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

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

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

$$5x+1=4+4\sqrt{2x+1}+2x+1$$

Add 4 and 1 to get 5.

$$5x+1=5+4\sqrt{2x+1}+2x$$

Subtract 5+2x from both sides of the equation.

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

To find the opposite of 5+2x, find the opposite of each term.

$$5x+1-5-2x=4\sqrt{2x+1}$$

Subtract 5 from 1 to get -4.

$$5x-4-2x=4\sqrt{2x+1}$$

Combine 5x and -2x to get 3x.

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

Square both sides of the equation.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-4\right)^{2}.

$$9x^{2}-24x+16=\left(4\sqrt{2x+1}\right)^{2}$$

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

$$9x^{2}-24x+16=4^{2}\left(\sqrt{2x+1}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$9x^{2}-24x+16=16\left(\sqrt{2x+1}\right)^{2}$$

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

$$9x^{2}-24x+16=16\left(2x+1\right)$$

Use the distributive property to multiply 16 by 2x+1.

$$9x^{2}-24x+16=32x+16$$

Subtract 32x from both sides.

$$9x^{2}-24x+16-32x=16$$

Combine -24x and -32x to get -56x.

$$9x^{2}-56x+16=16$$

Subtract 16 from both sides.

$$9x^{2}-56x+16-16=0$$

Subtract 16 from 16 to get 0.

$$9x^{2}-56x=0$$

Factor out x.

$$x\left(9x-56\right)=0$$

To find equation solutions, solve x=0 and 9x-56=0.

$$x=0 x=\frac{56}{9}$$

Substitute 0 for x in the equation \sqrt{5x+1}-\sqrt{2x+1}=2.

$$\sqrt{5\times 0+1}-\sqrt{2\times 0+1}=2$$

Simplify. The value x=0 does not satisfy the equation.

$$0=2$$

Substitute \frac{56}{9} for x in the equation \sqrt{5x+1}-\sqrt{2x+1}=2.

$$\sqrt{5\times \frac{56}{9}+1}-\sqrt{2\times \frac{56}{9}+1}=2$$

Simplify. The value x=\frac{56}{9} satisfies the equation.

$$2=2$$

Equation \sqrt{5x+1}=\sqrt{2x+1}+2 has a unique solution.

$$x=\frac{56}{9}$$