Solve x+3/5=sqrt{x^2+4}

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

Solve for x

$$x = \frac{91}{30} = 3\frac{1}{30} \approx 3.033333333$$

Solution Steps

Square both sides of the equation.

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

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

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\left(\sqrt{x^{2}+4}\right)^{2}$$

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

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=x^{2}+4$$

Subtract x^{2} from both sides.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}-x^{2}=4$$

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

$$\frac{6}{5}x+\frac{9}{25}=4$$

Subtract \frac{9}{25} from both sides.

$$\frac{6}{5}x=4-\frac{9}{25}$$

Subtract \frac{9}{25} from 4 to get \frac{91}{25}.

$$\frac{6}{5}x=\frac{91}{25}$$

Multiply both sides by \frac{5}{6}, the reciprocal of \frac{6}{5}.

$$x=\frac{91}{25}\times \frac{5}{6}$$

Multiply \frac{91}{25} and \frac{5}{6} to get \frac{91}{30}.

$$x=\frac{91}{30}$$

Substitute \frac{91}{30} for x in the equation x+\frac{3}{5}=\sqrt{x^{2}+4}.

$$\frac{91}{30}+\frac{3}{5}=\sqrt{\left(\frac{91}{30}\right)^{2}+4}$$

Simplify. The value x=\frac{91}{30} satisfies the equation.

$$\frac{109}{30}=\frac{109}{30}$$

Equation x+\frac{3}{5}=\sqrt{x^{2}+4} has a unique solution.

$$x=\frac{91}{30}$$