Solve sqrt{a+sqrt{3}}+sqrt{a-sqrt{3}}=9

$$\sqrt { a + \sqrt { 3 } } + \sqrt { a - \sqrt { 3 } } = 9$$

Solve for a

$$a = \frac{2191}{108} = 20\frac{31}{108} \approx 20.287037037$$

Solution Steps

Subtract \sqrt{a-\sqrt{3}} from both sides of the equation.

$$\sqrt{a+\sqrt{3}}=9-\sqrt{a-\sqrt{3}}$$

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(9-\sqrt{a-\sqrt{3}}\right)^{2}.

$$a+\sqrt{3}=81-18\sqrt{a-\sqrt{3}}+\left(\sqrt{a-\sqrt{3}}\right)^{2}$$

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

$$a+\sqrt{3}=81-18\sqrt{a-\sqrt{3}}+a-\sqrt{3}$$

Add 18\sqrt{a-\sqrt{3}} to both sides.

$$a+\sqrt{3}+18\sqrt{a-\sqrt{3}}=81+a-\sqrt{3}$$

Subtract a from both sides.

$$a+\sqrt{3}+18\sqrt{a-\sqrt{3}}-a=81-\sqrt{3}$$

Combine a and -a to get 0.

$$\sqrt{3}+18\sqrt{a-\sqrt{3}}=81-\sqrt{3}$$

Subtract \sqrt{3} from both sides.

$$18\sqrt{a-\sqrt{3}}=81-\sqrt{3}-\sqrt{3}$$

Combine -\sqrt{3} and -\sqrt{3} to get -2\sqrt{3}.

$$18\sqrt{a-\sqrt{3}}=81-2\sqrt{3}$$

Divide both sides by 18.

$$\frac{18\sqrt{a-\sqrt{3}}}{18}=\frac{81-2\sqrt{3}}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$\sqrt{a-\sqrt{3}}=\frac{81-2\sqrt{3}}{18}$$

Divide 81-2\sqrt{3} by 18.

$$\sqrt{a-\sqrt{3}}=-\frac{\sqrt{3}}{9}+\frac{9}{2}$$

Square both sides of the equation.

$$a-\sqrt{3}=\frac{2191}{108}-\sqrt{3}$$

Subtract -\sqrt{3} from both sides of the equation.

$$a-\sqrt{3}-\left(-\sqrt{3}\right)=\frac{2191}{108}-\sqrt{3}-\left(-\sqrt{3}\right)$$

Subtracting -\sqrt{3} from itself leaves 0.

$$a=\frac{2191}{108}-\sqrt{3}-\left(-\sqrt{3}\right)$$

Subtract -\sqrt{3} from \frac{2191}{108}-\sqrt{3}.

$$a=\frac{2191}{108}$$

Substitute \frac{2191}{108} for a in the equation \sqrt{a+\sqrt{3}}+\sqrt{a-\sqrt{3}}=9.

$$\sqrt{\frac{2191}{108}+\sqrt{3}}+\sqrt{\frac{2191}{108}-\sqrt{3}}=9$$

Simplify. The value a=\frac{2191}{108} satisfies the equation.

$$9=9$$

Equation \sqrt{a+\sqrt{3}}=-\sqrt{a-\sqrt{3}}+9 has a unique solution.

$$a=\frac{2191}{108}$$