Solve sqrt{9x-2}=sqrt{4x-3}+sqrt{x+1}

$$\sqrt { 9 x - 2 } = \sqrt { 4 x - 3 } + \sqrt { x + 1 }$$

Solve for x

$$x=3$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{9x-2}\right)^{2}=\left(\sqrt{4x-3}+\sqrt{x+1}\right)^{2}$$

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

$$9x-2=\left(\sqrt{4x-3}+\sqrt{x+1}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{4x-3}+\sqrt{x+1}\right)^{2}.

$$9x-2=\left(\sqrt{4x-3}\right)^{2}+2\sqrt{4x-3}\sqrt{x+1}+\left(\sqrt{x+1}\right)^{2}$$

Calculate \sqrt{4x-3} to the power of 2 and get 4x-3.

$$9x-2=4x-3+2\sqrt{4x-3}\sqrt{x+1}+\left(\sqrt{x+1}\right)^{2}$$

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

$$9x-2=4x-3+2\sqrt{4x-3}\sqrt{x+1}+x+1$$

Combine 4x and x to get 5x.

$$9x-2=5x-3+2\sqrt{4x-3}\sqrt{x+1}+1$$

Add -3 and 1 to get -2.

$$9x-2=5x-2+2\sqrt{4x-3}\sqrt{x+1}$$

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

$$9x-2-\left(5x-2\right)=2\sqrt{4x-3}\sqrt{x+1}$$

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

$$9x-2-5x+2=2\sqrt{4x-3}\sqrt{x+1}$$

Combine 9x and -5x to get 4x.

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

Add -2 and 2 to get 0.

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

Square both sides of the equation.

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

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

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

Calculate 4 to the power of 2 and get 16.

$$16x^{2}=\left(2\sqrt{4x-3}\sqrt{x+1}\right)^{2}$$

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

$$16x^{2}=2^{2}\left(\sqrt{4x-3}\right)^{2}\left(\sqrt{x+1}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$16x^{2}=4\left(\sqrt{4x-3}\right)^{2}\left(\sqrt{x+1}\right)^{2}$$

Calculate \sqrt{4x-3} to the power of 2 and get 4x-3.

$$16x^{2}=4\left(4x-3\right)\left(\sqrt{x+1}\right)^{2}$$

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

$$16x^{2}=4\left(4x-3\right)\left(x+1\right)$$

Use the distributive property to multiply 4 by 4x-3.

$$16x^{2}=\left(16x-12\right)\left(x+1\right)$$

Apply the distributive property by multiplying each term of 16x-12 by each term of x+1.

$$16x^{2}=16x^{2}+16x-12x-12$$

Combine 16x and -12x to get 4x.

$$16x^{2}=16x^{2}+4x-12$$

Subtract 16x^{2} from both sides.

$$16x^{2}-16x^{2}=4x-12$$

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

$$0=4x-12$$

Swap sides so that all variable terms are on the left hand side.

$$4x-12=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$4x=12$$

Divide both sides by 4.

$$x=\frac{12}{4}$$

Divide 12 by 4 to get 3.

$$x=3$$

Substitute 3 for x in the equation \sqrt{9x-2}=\sqrt{4x-3}+\sqrt{x+1}.

$$\sqrt{9\times 3-2}=\sqrt{4\times 3-3}+\sqrt{3+1}$$

Simplify. The value x=3 satisfies the equation.

$$5=5$$

Equation \sqrt{9x-2}=\sqrt{x+1}+\sqrt{4x-3} has a unique solution.

$$x=3$$