Solve sqrt{x}=x+2

$$\sqrt { x } = x + 2$$

Solve for x (complex solution)

$$x=\frac{-3+\sqrt{7}i}{2}\approx -1.5+1.322875656i$$

Solution Steps

Square both sides of the equation.

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

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

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

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

$$x=x^{2}+4x+4$$

Subtract x^{2} from both sides.

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

Subtract 4x from both sides.

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

Combine x and -4x to get -3x.

$$-3x-x^{2}=4$$

Subtract 4 from both sides.

$$-3x-x^{2}-4=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}-3x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -3 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+4\left(-4\right)}}{2\left(-1\right)}$$

Multiply 4 times -4.

$$x=\frac{-\left(-3\right)±\sqrt{9-16}}{2\left(-1\right)}$$

Add 9 to -16.

$$x=\frac{-\left(-3\right)±\sqrt{-7}}{2\left(-1\right)}$$

Take the square root of -7.

$$x=\frac{-\left(-3\right)±\sqrt{7}i}{2\left(-1\right)}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{7}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{3±\sqrt{7}i}{-2}$$

Now solve the equation x=\frac{3±\sqrt{7}i}{-2} when ± is plus. Add 3 to i\sqrt{7}.

$$x=\frac{3+\sqrt{7}i}{-2}$$

Divide 3+i\sqrt{7} by -2.

$$x=\frac{-\sqrt{7}i-3}{2}$$

Now solve the equation x=\frac{3±\sqrt{7}i}{-2} when ± is minus. Subtract i\sqrt{7} from 3.

$$x=\frac{-\sqrt{7}i+3}{-2}$$

Divide 3-i\sqrt{7} by -2.

$$x=\frac{-3+\sqrt{7}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{7}i-3}{2} x=\frac{-3+\sqrt{7}i}{2}$$

Substitute \frac{-\sqrt{7}i-3}{2} for x in the equation \sqrt{x}=x+2.

$$\sqrt{\frac{-\sqrt{7}i-3}{2}}=\frac{-\sqrt{7}i-3}{2}+2$$

Simplify. The value x=\frac{-\sqrt{7}i-3}{2} does not satisfy the equation.

$$-\left(\frac{1}{2}-\frac{1}{2}i\times 7^{\frac{1}{2}}\right)=-\frac{1}{2}i\times 7^{\frac{1}{2}}+\frac{1}{2}$$

Substitute \frac{-3+\sqrt{7}i}{2} for x in the equation \sqrt{x}=x+2.

$$\sqrt{\frac{-3+\sqrt{7}i}{2}}=\frac{-3+\sqrt{7}i}{2}+2$$

Simplify. The value x=\frac{-3+\sqrt{7}i}{2} satisfies the equation.

$$\frac{1}{2}+\frac{1}{2}i\times 7^{\frac{1}{2}}=\frac{1}{2}+\frac{1}{2}i\times 7^{\frac{1}{2}}$$

Equation \sqrt{x}=x+2 has a unique solution.

$$x=\frac{-3+\sqrt{7}i}{2}$$