Solve 2x+6=sqrt{x}

$$2x+6= \sqrt{ x }$$

Solve for x (complex solution)

$$x=\frac{-23+\sqrt{47}i}{8}\approx -2.875+0.856956825i$$

Solution Steps

Square both sides of the equation.

$$\left(2x+6\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+6\right)^{2}.

$$4x^{2}+24x+36=\left(\sqrt{x}\right)^{2}$$

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

$$4x^{2}+24x+36=x$$

Subtract x from both sides.

$$4x^{2}+24x+36-x=0$$

Combine 24x and -x to get 23x.

$$4x^{2}+23x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 23 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-23±\sqrt{23^{2}-4\times 4\times 36}}{2\times 4}$$

Square 23.

$$x=\frac{-23±\sqrt{529-4\times 4\times 36}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-23±\sqrt{529-16\times 36}}{2\times 4}$$

Multiply -16 times 36.

$$x=\frac{-23±\sqrt{529-576}}{2\times 4}$$

Add 529 to -576.

$$x=\frac{-23±\sqrt{-47}}{2\times 4}$$

Take the square root of -47.

$$x=\frac{-23±\sqrt{47}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-23±\sqrt{47}i}{8}$$

Now solve the equation x=\frac{-23±\sqrt{47}i}{8} when ± is plus. Add -23 to i\sqrt{47}.

$$x=\frac{-23+\sqrt{47}i}{8}$$

Now solve the equation x=\frac{-23±\sqrt{47}i}{8} when ± is minus. Subtract i\sqrt{47} from -23.

$$x=\frac{-\sqrt{47}i-23}{8}$$

The equation is now solved.

$$x=\frac{-23+\sqrt{47}i}{8} x=\frac{-\sqrt{47}i-23}{8}$$

Substitute \frac{-23+\sqrt{47}i}{8} for x in the equation 2x+6=\sqrt{x}.

$$2\times \frac{-23+\sqrt{47}i}{8}+6=\sqrt{\frac{-23+\sqrt{47}i}{8}}$$

Simplify. The value x=\frac{-23+\sqrt{47}i}{8} satisfies the equation.

$$\frac{1}{4}+\frac{1}{4}i\times 47^{\frac{1}{2}}=\frac{1}{4}+\frac{1}{4}i\times 47^{\frac{1}{2}}$$

Substitute \frac{-\sqrt{47}i-23}{8} for x in the equation 2x+6=\sqrt{x}.

$$2\times \frac{-\sqrt{47}i-23}{8}+6=\sqrt{\frac{-\sqrt{47}i-23}{8}}$$

Simplify. The value x=\frac{-\sqrt{47}i-23}{8} does not satisfy the equation.

$$-\frac{1}{4}i\times 47^{\frac{1}{2}}+\frac{1}{4}=-\left(\frac{1}{4}-\frac{1}{4}i\times 47^{\frac{1}{2}}\right)$$

Equation 2x+6=\sqrt{x} has a unique solution.

$$x=\frac{-23+\sqrt{47}i}{8}$$