Solve sqrt{3+a^2}+a=5

$$\sqrt { 3 + a ^ { 2 } } + a = 5$$

Solve for a

$$a = \frac{11}{5} = 2\frac{1}{5} = 2.2$$

Solution Steps

Subtract a from both sides of the equation.

$$\sqrt{3+a^{2}}=5-a$$

Square both sides of the equation.

$$\left(\sqrt{3+a^{2}}\right)^{2}=\left(5-a\right)^{2}$$

Calculate \sqrt{3+a^{2}} to the power of 2 and get 3+a^{2}.

$$3+a^{2}=\left(5-a\right)^{2}$$

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

$$3+a^{2}=25-10a+a^{2}$$

Add 10a to both sides.

$$3+a^{2}+10a=25+a^{2}$$

Subtract a^{2} from both sides.

$$3+a^{2}+10a-a^{2}=25$$

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

$$3+10a=25$$

Subtract 3 from both sides.

$$10a=25-3$$

Subtract 3 from 25 to get 22.

$$10a=22$$

Divide both sides by 10.

$$a=\frac{22}{10}$$

Reduce the fraction \frac{22}{10} to lowest terms by extracting and canceling out 2.

$$a=\frac{11}{5}$$

Substitute \frac{11}{5} for a in the equation \sqrt{3+a^{2}}+a=5.

$$\sqrt{3+\left(\frac{11}{5}\right)^{2}}+\frac{11}{5}=5$$

Simplify. The value a=\frac{11}{5} satisfies the equation.

$$5=5$$

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

$$a=\frac{11}{5}$$