Solve x+2/x+(2+3x)/2(4-3x)=1+4/3x^2-4x

$$\frac { x + 2 } { x } + \frac { ( 2 + 3 x ) } { 2 ( 4 - 3 x ) } = 1 + \frac { 4 } { 3 x ^ { 2 } - 4 x }$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{47}i}{3}\approx 1.666666667+2.2852182i$$

$$x=\frac{-\sqrt{47}i+5}{3}\approx 1.666666667-2.2852182i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,\frac{4}{3} since division by zero is not defined. Multiply both sides of the equation by 2x\left(3x-4\right), the least common multiple of x,2\left(4-3x\right),3x^{2}-4x.

$$\left(6x-8\right)\left(x+2\right)-x\left(2+3x\right)=2x\left(3x-4\right)+2\times 4$$

Use the distributive property to multiply 6x-8 by x+2 and combine like terms.

$$6x^{2}+4x-16-x\left(2+3x\right)=2x\left(3x-4\right)+2\times 4$$

Use the distributive property to multiply -x by 2+3x.

$$6x^{2}+4x-16-2x-3x^{2}=2x\left(3x-4\right)+2\times 4$$

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

$$6x^{2}+2x-16-3x^{2}=2x\left(3x-4\right)+2\times 4$$

Combine 6x^{2} and -3x^{2} to get 3x^{2}.

$$3x^{2}+2x-16=2x\left(3x-4\right)+2\times 4$$

Use the distributive property to multiply 2x by 3x-4.

$$3x^{2}+2x-16=6x^{2}-8x+2\times 4$$

Multiply 2 and 4 to get 8.

$$3x^{2}+2x-16=6x^{2}-8x+8$$

Subtract 6x^{2} from both sides.

$$3x^{2}+2x-16-6x^{2}=-8x+8$$

Combine 3x^{2} and -6x^{2} to get -3x^{2}.

$$-3x^{2}+2x-16=-8x+8$$

Add 8x to both sides.

$$-3x^{2}+2x-16+8x=8$$

Combine 2x and 8x to get 10x.

$$-3x^{2}+10x-16=8$$

Subtract 8 from both sides.

$$-3x^{2}+10x-16-8=0$$

Subtract 8 from -16 to get -24.

$$-3x^{2}+10x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 10 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-3\right)\left(-24\right)}}{2\left(-3\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-3\right)\left(-24\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-10±\sqrt{100+12\left(-24\right)}}{2\left(-3\right)}$$

Multiply 12 times -24.

$$x=\frac{-10±\sqrt{100-288}}{2\left(-3\right)}$$

Add 100 to -288.

$$x=\frac{-10±\sqrt{-188}}{2\left(-3\right)}$$

Take the square root of -188.

$$x=\frac{-10±2\sqrt{47}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-10±2\sqrt{47}i}{-6}$$

Now solve the equation x=\frac{-10±2\sqrt{47}i}{-6} when ± is plus. Add -10 to 2i\sqrt{47}.

$$x=\frac{-10+2\sqrt{47}i}{-6}$$

Divide -10+2i\sqrt{47} by -6.

$$x=\frac{-\sqrt{47}i+5}{3}$$

Now solve the equation x=\frac{-10±2\sqrt{47}i}{-6} when ± is minus. Subtract 2i\sqrt{47} from -10.

$$x=\frac{-2\sqrt{47}i-10}{-6}$$

Divide -10-2i\sqrt{47} by -6.

$$x=\frac{5+\sqrt{47}i}{3}$$

The equation is now solved.

$$x=\frac{-\sqrt{47}i+5}{3} x=\frac{5+\sqrt{47}i}{3}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,\frac{4}{3} since division by zero is not defined. Multiply both sides of the equation by 2x\left(3x-4\right), the least common multiple of x,2\left(4-3x\right),3x^{2}-4x.

$$\left(6x-8\right)\left(x+2\right)-x\left(2+3x\right)=2x\left(3x-4\right)+2\times 4$$

Use the distributive property to multiply 6x-8 by x+2 and combine like terms.

$$6x^{2}+4x-16-x\left(2+3x\right)=2x\left(3x-4\right)+2\times 4$$

Use the distributive property to multiply -x by 2+3x.

$$6x^{2}+4x-16-2x-3x^{2}=2x\left(3x-4\right)+2\times 4$$

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

$$6x^{2}+2x-16-3x^{2}=2x\left(3x-4\right)+2\times 4$$

Combine 6x^{2} and -3x^{2} to get 3x^{2}.

$$3x^{2}+2x-16=2x\left(3x-4\right)+2\times 4$$

Use the distributive property to multiply 2x by 3x-4.

$$3x^{2}+2x-16=6x^{2}-8x+2\times 4$$

Multiply 2 and 4 to get 8.

$$3x^{2}+2x-16=6x^{2}-8x+8$$

Subtract 6x^{2} from both sides.

$$3x^{2}+2x-16-6x^{2}=-8x+8$$

Combine 3x^{2} and -6x^{2} to get -3x^{2}.

$$-3x^{2}+2x-16=-8x+8$$

Add 8x to both sides.

$$-3x^{2}+2x-16+8x=8$$

Combine 2x and 8x to get 10x.

$$-3x^{2}+10x-16=8$$

Add 16 to both sides.

$$-3x^{2}+10x=8+16$$

Add 8 and 16 to get 24.

$$-3x^{2}+10x=24$$

Divide both sides by -3.

$$\frac{-3x^{2}+10x}{-3}=\frac{24}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{10}{-3}x=\frac{24}{-3}$$

Divide 10 by -3.

$$x^{2}-\frac{10}{3}x=\frac{24}{-3}$$

Divide 24 by -3.

$$x^{2}-\frac{10}{3}x=-8$$

Divide -\frac{10}{3}, the coefficient of the x term, by 2 to get -\frac{5}{3}. Then add the square of -\frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{10}{3}x+\left(-\frac{5}{3}\right)^{2}=-8+\left(-\frac{5}{3}\right)^{2}$$

Square -\frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=-8+\frac{25}{9}$$

Add -8 to \frac{25}{9}.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=-\frac{47}{9}$$

Factor x^{2}-\frac{10}{3}x+\frac{25}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{3}\right)^{2}=-\frac{47}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{3}\right)^{2}}=\sqrt{-\frac{47}{9}}$$

Simplify.

$$x-\frac{5}{3}=\frac{\sqrt{47}i}{3} x-\frac{5}{3}=-\frac{\sqrt{47}i}{3}$$

Add \frac{5}{3} to both sides of the equation.

$$x=\frac{5+\sqrt{47}i}{3} x=\frac{-\sqrt{47}i+5}{3}$$