Solve F(x)=2x+1/x+3x=-3

$$F ( x ) = \frac { 2 x + 1 } { x + 3 } x = - 3$$

Solve for x (complex solution)

$$x=\frac{\sqrt{14}i}{2}-1\approx -1+1.870828693i$$

$$x=-\frac{\sqrt{14}i}{2}-1\approx -1-1.870828693i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$\left(2x+1\right)x=-3\left(x+3\right)$$

Use the distributive property to multiply 2x+1 by x.

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

Use the distributive property to multiply -3 by x+3.

$$2x^{2}+x=-3x-9$$

Add 3x to both sides.

$$2x^{2}+x+3x=-9$$

Combine x and 3x to get 4x.

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

Add 9 to both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\times 9}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\times 9}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\times 9}}{2\times 2}$$

Multiply -8 times 9.

$$x=\frac{-4±\sqrt{16-72}}{2\times 2}$$

Add 16 to -72.

$$x=\frac{-4±\sqrt{-56}}{2\times 2}$$

Take the square root of -56.

$$x=\frac{-4±2\sqrt{14}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±2\sqrt{14}i}{4}$$

Now solve the equation x=\frac{-4±2\sqrt{14}i}{4} when ± is plus. Add -4 to 2i\sqrt{14}.

$$x=\frac{-4+2\sqrt{14}i}{4}$$

Divide -4+2i\sqrt{14} by 4.

$$x=\frac{\sqrt{14}i}{2}-1$$

Now solve the equation x=\frac{-4±2\sqrt{14}i}{4} when ± is minus. Subtract 2i\sqrt{14} from -4.

$$x=\frac{-2\sqrt{14}i-4}{4}$$

Divide -4-2i\sqrt{14} by 4.

$$x=-\frac{\sqrt{14}i}{2}-1$$

The equation is now solved.

$$x=\frac{\sqrt{14}i}{2}-1 x=-\frac{\sqrt{14}i}{2}-1$$

Steps for Completing the Square

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$\left(2x+1\right)x=-3\left(x+3\right)$$

Use the distributive property to multiply 2x+1 by x.

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

Use the distributive property to multiply -3 by x+3.

$$2x^{2}+x=-3x-9$$

Add 3x to both sides.

$$2x^{2}+x+3x=-9$$

Combine x and 3x to get 4x.

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

Divide both sides by 2.

$$\frac{2x^{2}+4x}{2}=-\frac{9}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{4}{2}x=-\frac{9}{2}$$

Divide 4 by 2.

$$x^{2}+2x=-\frac{9}{2}$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=-\frac{9}{2}+1^{2}$$

Square 1.

$$x^{2}+2x+1=-\frac{9}{2}+1$$

Add -\frac{9}{2} to 1.

$$x^{2}+2x+1=-\frac{7}{2}$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=\frac{\sqrt{14}i}{2} x+1=-\frac{\sqrt{14}i}{2}$$

Subtract 1 from both sides of the equation.

$$x=\frac{\sqrt{14}i}{2}-1 x=-\frac{\sqrt{14}i}{2}-1$$