Solve x^2+sqrt{x^2-18}=18

$$x ^ { 2 } + \sqrt { x ^ { 2 } - 18 } = 18$$

Solve for x

$$x=3\sqrt{2}\approx 4.242640687$$

$$x=-3\sqrt{2}\approx -4.242640687$$

Solution Steps

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

$$\sqrt{x^{2}-18}=18-x^{2}$$

Square both sides of the equation.

$$\left(\sqrt{x^{2}-18}\right)^{2}=\left(18-x^{2}\right)^{2}$$

Calculate \sqrt{x^{2}-18} to the power of 2 and get x^{2}-18.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(18-x^{2}\right)^{2}.

$$x^{2}-18=324-36x^{2}+\left(x^{2}\right)^{2}$$

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

$$x^{2}-18=324-36x^{2}+x^{4}$$

Subtract 324 from both sides.

$$x^{2}-18-324=-36x^{2}+x^{4}$$

Subtract 324 from -18 to get -342.

$$x^{2}-342=-36x^{2}+x^{4}$$

Add 36x^{2} to both sides.

$$x^{2}-342+36x^{2}=x^{4}$$

Combine x^{2} and 36x^{2} to get 37x^{2}.

$$37x^{2}-342=x^{4}$$

Subtract x^{4} from both sides.

$$37x^{2}-342-x^{4}=0$$

Substitute t for x^{2}.

$$-t^{2}+37t-342=0$$

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, 37 for b, and -342 for c in the quadratic formula.

$$t=\frac{-37±\sqrt{37^{2}-4\left(-1\right)\left(-342\right)}}{-2}$$

Do the calculations.

$$t=\frac{-37±1}{-2}$$

Solve the equation t=\frac{-37±1}{-2} when ± is plus and when ± is minus.

$$t=18 t=19$$

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

$$x=3\sqrt{2} x=-3\sqrt{2} x=\sqrt{19} x=-\sqrt{19}$$

Substitute 3\sqrt{2} for x in the equation x^{2}+\sqrt{x^{2}-18}=18.

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

Simplify. The value x=3\sqrt{2} satisfies the equation.

$$18=18$$

Substitute -3\sqrt{2} for x in the equation x^{2}+\sqrt{x^{2}-18}=18.

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

Simplify. The value x=-3\sqrt{2} satisfies the equation.

$$18=18$$

Substitute \sqrt{19} for x in the equation x^{2}+\sqrt{x^{2}-18}=18.

$$\left(\sqrt{19}\right)^{2}+\sqrt{\left(\sqrt{19}\right)^{2}-18}=18$$

Simplify. The value x=\sqrt{19} does not satisfy the equation.

$$20=18$$

Substitute -\sqrt{19} for x in the equation x^{2}+\sqrt{x^{2}-18}=18.

$$\left(-\sqrt{19}\right)^{2}+\sqrt{\left(-\sqrt{19}\right)^{2}-18}=18$$

Simplify. The value x=-\sqrt{19} does not satisfy the equation.

$$20=18$$

List all solutions of \sqrt{x^{2}-18}=18-x^{2}.

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