Solve 4+x=2sqrt{x^2+4}

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

Solve for x

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

$$x=0$$

Solution Steps

Square both sides of the equation.

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

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

$$16+8x+x^{2}=\left(2\sqrt{x^{2}+4}\right)^{2}$$

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

$$16+8x+x^{2}=2^{2}\left(\sqrt{x^{2}+4}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$16+8x+x^{2}=4\left(\sqrt{x^{2}+4}\right)^{2}$$

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

$$16+8x+x^{2}=4\left(x^{2}+4\right)$$

Use the distributive property to multiply 4 by x^{2}+4.

$$16+8x+x^{2}=4x^{2}+16$$

Subtract 4x^{2} from both sides.

$$16+8x+x^{2}-4x^{2}=16$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$16+8x-3x^{2}=16$$

Subtract 16 from both sides.

$$16+8x-3x^{2}-16=0$$

Subtract 16 from 16 to get 0.

$$8x-3x^{2}=0$$

Factor out x.

$$x\left(8-3x\right)=0$$

To find equation solutions, solve x=0 and 8-3x=0.

$$x=0 x=\frac{8}{3}$$

Substitute 0 for x in the equation 4+x=2\sqrt{x^{2}+4}.

$$4+0=2\sqrt{0^{2}+4}$$

Simplify. The value x=0 satisfies the equation.

$$4=4$$

Substitute \frac{8}{3} for x in the equation 4+x=2\sqrt{x^{2}+4}.

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

Simplify. The value x=\frac{8}{3} satisfies the equation.

$$\frac{20}{3}=\frac{20}{3}$$

List all solutions of x+4=2\sqrt{x^{2}+4}.

$$x=0 x=\frac{8}{3}$$