Solve 2x+3=sqrt{x-2}

$$2x+3= \sqrt{ x-2 }$$

Solve for x (complex solution)

$$x=\frac{-11+\sqrt{55}i}{8}\approx -1.375+0.927024811i$$

Solution Steps

Square both sides of the equation.

$$\left(2x+3\right)^{2}=\left(\sqrt{x-2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$4x^{2}+12x+9=\left(\sqrt{x-2}\right)^{2}$$

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

$$4x^{2}+12x+9=x-2$$

Subtract x from both sides.

$$4x^{2}+12x+9-x=-2$$

Combine 12x and -x to get 11x.

$$4x^{2}+11x+9=-2$$

Add 2 to both sides.

$$4x^{2}+11x+9+2=0$$

Add 9 and 2 to get 11.

$$4x^{2}+11x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 11 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 4\times 11}}{2\times 4}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 4\times 11}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-11±\sqrt{121-16\times 11}}{2\times 4}$$

Multiply -16 times 11.

$$x=\frac{-11±\sqrt{121-176}}{2\times 4}$$

Add 121 to -176.

$$x=\frac{-11±\sqrt{-55}}{2\times 4}$$

Take the square root of -55.

$$x=\frac{-11±\sqrt{55}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-11±\sqrt{55}i}{8}$$

Now solve the equation x=\frac{-11±\sqrt{55}i}{8} when ± is plus. Add -11 to i\sqrt{55}.

$$x=\frac{-11+\sqrt{55}i}{8}$$

Now solve the equation x=\frac{-11±\sqrt{55}i}{8} when ± is minus. Subtract i\sqrt{55} from -11.

$$x=\frac{-\sqrt{55}i-11}{8}$$

The equation is now solved.

$$x=\frac{-11+\sqrt{55}i}{8} x=\frac{-\sqrt{55}i-11}{8}$$

Substitute \frac{-11+\sqrt{55}i}{8} for x in the equation 2x+3=\sqrt{x-2}.

$$2\times \frac{-11+\sqrt{55}i}{8}+3=\sqrt{\frac{-11+\sqrt{55}i}{8}-2}$$

Simplify. The value x=\frac{-11+\sqrt{55}i}{8} satisfies the equation.

$$\frac{1}{4}+\frac{1}{4}i\times 55^{\frac{1}{2}}=\frac{1}{4}+\frac{1}{4}i\times 55^{\frac{1}{2}}$$

Substitute \frac{-\sqrt{55}i-11}{8} for x in the equation 2x+3=\sqrt{x-2}.

$$2\times \frac{-\sqrt{55}i-11}{8}+3=\sqrt{\frac{-\sqrt{55}i-11}{8}-2}$$

Simplify. The value x=\frac{-\sqrt{55}i-11}{8} does not satisfy the equation.

$$-\frac{1}{4}i\times 55^{\frac{1}{2}}+\frac{1}{4}=-\left(\frac{1}{4}-\frac{1}{4}i\times 55^{\frac{1}{2}}\right)$$

Equation 2x+3=\sqrt{x-2} has a unique solution.

$$x=\frac{-11+\sqrt{55}i}{8}$$