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

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

Solve for x

$$x=4$$

$$x=0$$

Solution Steps

Square both sides of the equation.

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

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

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

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

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

Add 4 and 1 to get 5.

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

Calculate \sqrt{4x+9} to the power of 2 and get 4x+9.

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

Subtract 3x+5 from both sides of the equation.

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

To find the opposite of 3x+5, find the opposite of each term.

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

Combine 4x and -3x to get x.

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

Subtract 5 from 9 to get 4.

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

Square both sides of the equation.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

$$12x+16=\left(x+4\right)^{2}$$

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

$$12x+16=x^{2}+8x+16$$

Subtract x^{2} from both sides.

$$12x+16-x^{2}=8x+16$$

Subtract 8x from both sides.

$$12x+16-x^{2}-8x=16$$

Combine 12x and -8x to get 4x.

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

Subtract 16 from both sides.

$$4x+16-x^{2}-16=0$$

Subtract 16 from 16 to get 0.

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

Factor out x.

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

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

$$x=0 x=4$$

Substitute 0 for x in the equation \sqrt{3x+4}+1=\sqrt{4x+9}.

$$\sqrt{3\times 0+4}+1=\sqrt{4\times 0+9}$$

Simplify. The value x=0 satisfies the equation.

$$3=3$$

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

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

Simplify. The value x=4 satisfies the equation.

$$5=5$$

List all solutions of \sqrt{3x+4}+1=\sqrt{4x+9}.

$$x=0 x=4$$