Solve 0.5x+3=2sqrt{x-2}

$$0.5x+3=2 \sqrt{ x-2 }$$

Solve for x (complex solution)

$$x=2+8i$$

$$x=2-8i$$

Solution Steps

Square both sides of the equation.

$$\left(0.5x+3\right)^{2}=\left(2\sqrt{x-2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(0.5x+3\right)^{2}.

$$0.25x^{2}+3x+9=\left(2\sqrt{x-2}\right)^{2}$$

Expand \left(2\sqrt{x-2}\right)^{2}.

$$0.25x^{2}+3x+9=2^{2}\left(\sqrt{x-2}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$0.25x^{2}+3x+9=4\left(\sqrt{x-2}\right)^{2}$$

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

$$0.25x^{2}+3x+9=4\left(x-2\right)$$

Use the distributive property to multiply 4 by x-2.

$$0.25x^{2}+3x+9=4x-8$$

Subtract 4x from both sides.

$$0.25x^{2}+3x+9-4x=-8$$

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

$$0.25x^{2}-x+9=-8$$

Add 8 to both sides.

$$0.25x^{2}-x+9+8=0$$

Add 9 and 8 to get 17.

$$0.25x^{2}-x+17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.25 for a, -1 for b, and 17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 0.25\times 17}}{2\times 0.25}$$

Multiply -4 times 0.25.

$$x=\frac{-\left(-1\right)±\sqrt{1-17}}{2\times 0.25}$$

Add 1 to -17.

$$x=\frac{-\left(-1\right)±\sqrt{-16}}{2\times 0.25}$$

Take the square root of -16.

$$x=\frac{-\left(-1\right)±4i}{2\times 0.25}$$

The opposite of -1 is 1.

$$x=\frac{1±4i}{2\times 0.25}$$

Multiply 2 times 0.25.

$$x=\frac{1±4i}{0.5}$$

Now solve the equation x=\frac{1±4i}{0.5} when ± is plus. Add 1 to 4i.

$$x=\frac{1+4i}{0.5}$$

Divide 1+4i by 0.5 by multiplying 1+4i by the reciprocal of 0.5.

$$x=2+8i$$

Now solve the equation x=\frac{1±4i}{0.5} when ± is minus. Subtract 4i from 1.

$$x=\frac{1-4i}{0.5}$$

Divide 1-4i by 0.5 by multiplying 1-4i by the reciprocal of 0.5.

$$x=2-8i$$

The equation is now solved.

$$x=2+8i x=2-8i$$

Substitute 2+8i for x in the equation 0.5x+3=2\sqrt{x-2}.

$$0.5\left(2+8i\right)+3=2\sqrt{2+8i-2}$$

Simplify. The value x=2+8i satisfies the equation.

$$4+4i=4+4i$$

Substitute 2-8i for x in the equation 0.5x+3=2\sqrt{x-2}.

$$0.5\left(2-8i\right)+3=2\sqrt{2-8i-2}$$

Simplify. The value x=2-8i satisfies the equation.

$$4-4i=4-4i$$

List all solutions of \frac{x}{2}+3=2\sqrt{x-2}.

$$x=2+8i x=2-8i$$