Solve 1-sqrt{9+x^2}=-x

$$1 - \sqrt { 9 + x ^ { 2 } } = - x$$

Solve for x

$$x=4$$

Solution Steps

Subtract 1 from both sides of the equation.

$$-\sqrt{9+x^{2}}=-x-1$$

Square both sides of the equation.

$$\left(-\sqrt{9+x^{2}}\right)^{2}=\left(-x-1\right)^{2}$$

Expand \left(-\sqrt{9+x^{2}}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{9+x^{2}}\right)^{2}=\left(-x-1\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{9+x^{2}}\right)^{2}=\left(-x-1\right)^{2}$$

Calculate \sqrt{9+x^{2}} to the power of 2 and get 9+x^{2}.

$$1\left(9+x^{2}\right)=\left(-x-1\right)^{2}$$

Use the distributive property to multiply 1 by 9+x^{2}.

$$9+x^{2}=\left(-x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-x-1\right)^{2}.

$$9+x^{2}=\left(-x\right)^{2}-2\left(-x\right)+1$$

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

$$9+x^{2}=x^{2}-2\left(-x\right)+1$$

Multiply -2 and -1 to get 2.

$$9+x^{2}=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$9+x^{2}-x^{2}=2x+1$$

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

$$9=2x+1$$

Swap sides so that all variable terms are on the left hand side.

$$2x+1=9$$

Subtract 1 from both sides.

$$2x=9-1$$

Subtract 1 from 9 to get 8.

$$2x=8$$

Divide both sides by 2.

$$x=\frac{8}{2}$$

Divide 8 by 2 to get 4.

$$x=4$$

Substitute 4 for x in the equation 1-\sqrt{9+x^{2}}=-x.

$$1-\sqrt{9+4^{2}}=-4$$

Simplify. The value x=4 satisfies the equation.

$$-4=-4$$

Equation -\sqrt{x^{2}+9}=-x-1 has a unique solution.

$$x=4$$