Solve x=sqrt{x}-10

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

Solve for x (complex solution)

$$x=\frac{-19+\sqrt{39}i}{2}\approx -9.5+3.122498999i$$

Solution Steps

Subtract -10 from both sides of the equation.

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

Square both sides of the equation.

$$\left(x+10\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+10\right)^{2}.

$$x^{2}+20x+100=\left(\sqrt{x}\right)^{2}$$

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

$$x^{2}+20x+100=x$$

Subtract x from both sides.

$$x^{2}+20x+100-x=0$$

Combine 20x and -x to get 19x.

$$x^{2}+19x+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 19 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\times 100}}{2}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\times 100}}{2}$$

Multiply -4 times 100.

$$x=\frac{-19±\sqrt{361-400}}{2}$$

Add 361 to -400.

$$x=\frac{-19±\sqrt{-39}}{2}$$

Take the square root of -39.

$$x=\frac{-19±\sqrt{39}i}{2}$$

Now solve the equation x=\frac{-19±\sqrt{39}i}{2} when ± is plus. Add -19 to i\sqrt{39}.

$$x=\frac{-19+\sqrt{39}i}{2}$$

Now solve the equation x=\frac{-19±\sqrt{39}i}{2} when ± is minus. Subtract i\sqrt{39} from -19.

$$x=\frac{-\sqrt{39}i-19}{2}$$

The equation is now solved.

$$x=\frac{-19+\sqrt{39}i}{2} x=\frac{-\sqrt{39}i-19}{2}$$

Substitute \frac{-19+\sqrt{39}i}{2} for x in the equation x=\sqrt{x}-10.

$$\frac{-19+\sqrt{39}i}{2}=\sqrt{\frac{-19+\sqrt{39}i}{2}}-10$$

Simplify. The value x=\frac{-19+\sqrt{39}i}{2} satisfies the equation.

$$-\frac{19}{2}+\frac{1}{2}i\times 39^{\frac{1}{2}}=-\frac{19}{2}+\frac{1}{2}i\times 39^{\frac{1}{2}}$$

Substitute \frac{-\sqrt{39}i-19}{2} for x in the equation x=\sqrt{x}-10.

$$\frac{-\sqrt{39}i-19}{2}=\sqrt{\frac{-\sqrt{39}i-19}{2}}-10$$

Simplify. The value x=\frac{-\sqrt{39}i-19}{2} does not satisfy the equation.

$$-\frac{1}{2}i\times 39^{\frac{1}{2}}-\frac{19}{2}=-\frac{21}{2}+\frac{1}{2}i\times 39^{\frac{1}{2}}$$

Equation x+10=\sqrt{x} has a unique solution.

$$x=\frac{-19+\sqrt{39}i}{2}$$