Solve x+29=sqrt{x}

$$x + 29 = \sqrt { x }$$

Solve for x (complex solution)

$$x=\frac{-57+\sqrt{115}i}{2}\approx -28.5+5.361902647i$$

Solution Steps

Square both sides of the equation.

$$\left(x+29\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+29\right)^{2}.

$$x^{2}+58x+841=\left(\sqrt{x}\right)^{2}$$

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

$$x^{2}+58x+841=x$$

Subtract x from both sides.

$$x^{2}+58x+841-x=0$$

Combine 58x and -x to get 57x.

$$x^{2}+57x+841=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 57 for b, and 841 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-57±\sqrt{57^{2}-4\times 841}}{2}$$

Square 57.

$$x=\frac{-57±\sqrt{3249-4\times 841}}{2}$$

Multiply -4 times 841.

$$x=\frac{-57±\sqrt{3249-3364}}{2}$$

Add 3249 to -3364.

$$x=\frac{-57±\sqrt{-115}}{2}$$

Take the square root of -115.

$$x=\frac{-57±\sqrt{115}i}{2}$$

Now solve the equation x=\frac{-57±\sqrt{115}i}{2} when ± is plus. Add -57 to i\sqrt{115}.

$$x=\frac{-57+\sqrt{115}i}{2}$$

Now solve the equation x=\frac{-57±\sqrt{115}i}{2} when ± is minus. Subtract i\sqrt{115} from -57.

$$x=\frac{-\sqrt{115}i-57}{2}$$

The equation is now solved.

$$x=\frac{-57+\sqrt{115}i}{2} x=\frac{-\sqrt{115}i-57}{2}$$

Substitute \frac{-57+\sqrt{115}i}{2} for x in the equation x+29=\sqrt{x}.

$$\frac{-57+\sqrt{115}i}{2}+29=\sqrt{\frac{-57+\sqrt{115}i}{2}}$$

Simplify. The value x=\frac{-57+\sqrt{115}i}{2} satisfies the equation.

$$\frac{1}{2}+\frac{1}{2}i\times 115^{\frac{1}{2}}=\frac{1}{2}+\frac{1}{2}i\times 115^{\frac{1}{2}}$$

Substitute \frac{-\sqrt{115}i-57}{2} for x in the equation x+29=\sqrt{x}.

$$\frac{-\sqrt{115}i-57}{2}+29=\sqrt{\frac{-\sqrt{115}i-57}{2}}$$

Simplify. The value x=\frac{-\sqrt{115}i-57}{2} does not satisfy the equation.

$$-\frac{1}{2}i\times 115^{\frac{1}{2}}+\frac{1}{2}=-\left(\frac{1}{2}-\frac{1}{2}i\times 115^{\frac{1}{2}}\right)$$

Equation x+29=\sqrt{x} has a unique solution.

$$x=\frac{-57+\sqrt{115}i}{2}$$