Solve x^2-9=sqrt{11-x^2}

$${ x }^{ 2 } -9= \sqrt{ 11- { x }^{ 2 } }$$

Solve for x

$$x=\sqrt{10}\approx 3.16227766$$

$$x=-\sqrt{10}\approx -3.16227766$$

Solution Steps

Square both sides of the equation.

$$\left(x^{2}-9\right)^{2}=\left(\sqrt{11-x^{2}}\right)^{2}$$

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

$$\left(x^{2}\right)^{2}-18x^{2}+81=\left(\sqrt{11-x^{2}}\right)^{2}$$

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

$$x^{4}-18x^{2}+81=\left(\sqrt{11-x^{2}}\right)^{2}$$

Calculate \sqrt{11-x^{2}} to the power of 2 and get 11-x^{2}.

$$x^{4}-18x^{2}+81=11-x^{2}$$

Subtract 11 from both sides.

$$x^{4}-18x^{2}+81-11=-x^{2}$$

Subtract 11 from 81 to get 70.

$$x^{4}-18x^{2}+70=-x^{2}$$

Add x^{2} to both sides.

$$x^{4}-18x^{2}+70+x^{2}=0$$

Combine -18x^{2} and x^{2} to get -17x^{2}.

$$x^{4}-17x^{2}+70=0$$

Substitute t for x^{2}.

$$t^{2}-17t+70=0$$

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, -17 for b, and 70 for c in the quadratic formula.

$$t=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 1\times 70}}{2}$$

Do the calculations.

$$t=\frac{17±3}{2}$$

Solve the equation t=\frac{17±3}{2} when ± is plus and when ± is minus.

$$t=10 t=7$$

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

$$x=\sqrt{10} x=-\sqrt{10} x=\sqrt{7} x=-\sqrt{7}$$

Substitute \sqrt{10} for x in the equation x^{2}-9=\sqrt{11-x^{2}}.

$$\left(\sqrt{10}\right)^{2}-9=\sqrt{11-\left(\sqrt{10}\right)^{2}}$$

Simplify. The value x=\sqrt{10} satisfies the equation.

$$1=1$$

Substitute -\sqrt{10} for x in the equation x^{2}-9=\sqrt{11-x^{2}}.

$$\left(-\sqrt{10}\right)^{2}-9=\sqrt{11-\left(-\sqrt{10}\right)^{2}}$$

Simplify. The value x=-\sqrt{10} satisfies the equation.

$$1=1$$

Substitute \sqrt{7} for x in the equation x^{2}-9=\sqrt{11-x^{2}}.

$$\left(\sqrt{7}\right)^{2}-9=\sqrt{11-\left(\sqrt{7}\right)^{2}}$$

Simplify. The value x=\sqrt{7} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=2$$

Substitute -\sqrt{7} for x in the equation x^{2}-9=\sqrt{11-x^{2}}.

$$\left(-\sqrt{7}\right)^{2}-9=\sqrt{11-\left(-\sqrt{7}\right)^{2}}$$

Simplify. The value x=-\sqrt{7} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=2$$

List all solutions of x^{2}-9=\sqrt{11-x^{2}}.

$$x=\sqrt{10} x=-\sqrt{10}$$