Solve x^2+sqrt{x^2-3}=9

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

Solve for x

$$x=\sqrt{7}\approx 2.645751311$$

$$x=-\sqrt{7}\approx -2.645751311$$

Solution Steps

Subtract x^{2} from both sides of the equation.

$$\sqrt{x^{2}-3}=9-x^{2}$$

Square both sides of the equation.

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

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

$$x^{2}-3=\left(9-x^{2}\right)^{2}$$

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{2}-3=81-18x^{2}+x^{4}$$

Subtract 81 from both sides.

$$x^{2}-3-81=-18x^{2}+x^{4}$$

Subtract 81 from -3 to get -84.

$$x^{2}-84=-18x^{2}+x^{4}$$

Add 18x^{2} to both sides.

$$x^{2}-84+18x^{2}=x^{4}$$

Combine x^{2} and 18x^{2} to get 19x^{2}.

$$19x^{2}-84=x^{4}$$

Subtract x^{4} from both sides.

$$19x^{2}-84-x^{4}=0$$

Substitute t for x^{2}.

$$-t^{2}+19t-84=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute -1 for a, 19 for b, and -84 for c in the quadratic formula.

$$t=\frac{-19±\sqrt{19^{2}-4\left(-1\right)\left(-84\right)}}{-2}$$

Do the calculations.

$$t=\frac{-19±5}{-2}$$

Solve the equation t=\frac{-19±5}{-2} when ± is plus and when ± is minus.

$$t=7 t=12$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=\sqrt{7} x=-\sqrt{7} x=2\sqrt{3} x=-2\sqrt{3}$$

Substitute \sqrt{7} for x in the equation x^{2}+\sqrt{x^{2}-3}=9.

$$\left(\sqrt{7}\right)^{2}+\sqrt{\left(\sqrt{7}\right)^{2}-3}=9$$

Simplify. The value x=\sqrt{7} satisfies the equation.

$$9=9$$

Substitute -\sqrt{7} for x in the equation x^{2}+\sqrt{x^{2}-3}=9.

$$\left(-\sqrt{7}\right)^{2}+\sqrt{\left(-\sqrt{7}\right)^{2}-3}=9$$

Simplify. The value x=-\sqrt{7} satisfies the equation.

$$9=9$$

Substitute 2\sqrt{3} for x in the equation x^{2}+\sqrt{x^{2}-3}=9.

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

Simplify. The value x=2\sqrt{3} does not satisfy the equation.

$$15=9$$

Substitute -2\sqrt{3} for x in the equation x^{2}+\sqrt{x^{2}-3}=9.

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

Simplify. The value x=-2\sqrt{3} does not satisfy the equation.

$$15=9$$

List all solutions of \sqrt{x^{2}-3}=9-x^{2}.

$$x=\sqrt{7} x=-\sqrt{7}$$