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

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

Solve for x (complex solution)

$$x=-1+\sqrt{6}i\approx -1+2.449489743i$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{2x-3} to the power of 2 and get 2x-3.

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

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

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

Subtract x^{2} from both sides.

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

Subtract 4x from both sides.

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

Combine 2x and -4x to get -2x.

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

Subtract 4 from both sides.

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

Subtract 4 from -3 to get -7.

$$-2x-7-x^{2}=0$$

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.

$$-x^{2}-2x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -2 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\left(-7\right)}}{2\left(-1\right)}$$

Multiply 4 times -7.

$$x=\frac{-\left(-2\right)±\sqrt{4-28}}{2\left(-1\right)}$$

Add 4 to -28.

$$x=\frac{-\left(-2\right)±\sqrt{-24}}{2\left(-1\right)}$$

Take the square root of -24.

$$x=\frac{-\left(-2\right)±2\sqrt{6}i}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{6}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±2\sqrt{6}i}{-2}$$

Now solve the equation x=\frac{2±2\sqrt{6}i}{-2} when ± is plus. Add 2 to 2i\sqrt{6}.

$$x=\frac{2+2\sqrt{6}i}{-2}$$

Divide 2+2i\sqrt{6} by -2.

$$x=-\sqrt{6}i-1$$

Now solve the equation x=\frac{2±2\sqrt{6}i}{-2} when ± is minus. Subtract 2i\sqrt{6} from 2.

$$x=\frac{-2\sqrt{6}i+2}{-2}$$

Divide 2-2i\sqrt{6} by -2.

$$x=-1+\sqrt{6}i$$

The equation is now solved.

$$x=-\sqrt{6}i-1 x=-1+\sqrt{6}i$$

Substitute -\sqrt{6}i-1 for x in the equation \sqrt{2x-3}=x+2.

$$\sqrt{2\left(-\sqrt{6}i-1\right)-3}=-\sqrt{6}i-1+2$$

Simplify. The value x=-\sqrt{6}i-1 does not satisfy the equation.

$$-\left(1-i\times 6^{\frac{1}{2}}\right)=-i\times 6^{\frac{1}{2}}+1$$

Substitute -1+\sqrt{6}i for x in the equation \sqrt{2x-3}=x+2.

$$\sqrt{2\left(-1+\sqrt{6}i\right)-3}=-1+\sqrt{6}i+2$$

Simplify. The value x=-1+\sqrt{6}i satisfies the equation.

$$1+i\times 6^{\frac{1}{2}}=1+i\times 6^{\frac{1}{2}}$$

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

$$x=-1+\sqrt{6}i$$