Solve 340/x-5=300/x+5-2/3

$$\frac{ 340 }{ x-5 } = \frac{ 300 }{ x+5 } - \frac{ 2 }{ 3 }$$

Solve for x (complex solution)

$$x=-30+5\sqrt{155}i\approx -30+62.24949799i$$

$$x=-5\sqrt{155}i-30\approx -30-62.24949799i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-5\right)\left(x+5\right), the least common multiple of x-5,x+5,3.

$$\left(3x+15\right)\times 340=\left(3x-15\right)\times 300+3\left(x-5\right)\left(x+5\right)\left(-\frac{2}{3}\right)$$

Use the distributive property to multiply 3x+15 by 340.

$$1020x+5100=\left(3x-15\right)\times 300+3\left(x-5\right)\left(x+5\right)\left(-\frac{2}{3}\right)$$

Use the distributive property to multiply 3x-15 by 300.

$$1020x+5100=900x-4500+3\left(x-5\right)\left(x+5\right)\left(-\frac{2}{3}\right)$$

Multiply 3 and -\frac{2}{3} to get -2.

$$1020x+5100=900x-4500-2\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply -2 by x-5.

$$1020x+5100=900x-4500+\left(-2x+10\right)\left(x+5\right)$$

Use the distributive property to multiply -2x+10 by x+5 and combine like terms.

$$1020x+5100=900x-4500-2x^{2}+50$$

Add -4500 and 50 to get -4450.

$$1020x+5100=900x-4450-2x^{2}$$

Subtract 900x from both sides.

$$1020x+5100-900x=-4450-2x^{2}$$

Combine 1020x and -900x to get 120x.

$$120x+5100=-4450-2x^{2}$$

Subtract -4450 from both sides.

$$120x+5100-\left(-4450\right)=-2x^{2}$$

The opposite of -4450 is 4450.

$$120x+5100+4450=-2x^{2}$$

Add 2x^{2} to both sides.

$$120x+5100+4450+2x^{2}=0$$

Add 5100 and 4450 to get 9550.

$$120x+9550+2x^{2}=0$$

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.

$$2x^{2}+120x+9550=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 120 for b, and 9550 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-120±\sqrt{120^{2}-4\times 2\times 9550}}{2\times 2}$$

Square 120.

$$x=\frac{-120±\sqrt{14400-4\times 2\times 9550}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-120±\sqrt{14400-8\times 9550}}{2\times 2}$$

Multiply -8 times 9550.

$$x=\frac{-120±\sqrt{14400-76400}}{2\times 2}$$

Add 14400 to -76400.

$$x=\frac{-120±\sqrt{-62000}}{2\times 2}$$

Take the square root of -62000.

$$x=\frac{-120±20\sqrt{155}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-120±20\sqrt{155}i}{4}$$

Now solve the equation x=\frac{-120±20\sqrt{155}i}{4} when ± is plus. Add -120 to 20i\sqrt{155}.

$$x=\frac{-120+20\sqrt{155}i}{4}$$

Divide -120+20i\sqrt{155} by 4.

$$x=-30+5\sqrt{155}i$$

Now solve the equation x=\frac{-120±20\sqrt{155}i}{4} when ± is minus. Subtract 20i\sqrt{155} from -120.

$$x=\frac{-20\sqrt{155}i-120}{4}$$

Divide -120-20i\sqrt{155} by 4.

$$x=-5\sqrt{155}i-30$$

The equation is now solved.

$$x=-30+5\sqrt{155}i x=-5\sqrt{155}i-30$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-5\right)\left(x+5\right), the least common multiple of x-5,x+5,3.

$$\left(3x+15\right)\times 340=\left(3x-15\right)\times 300+3\left(x-5\right)\left(x+5\right)\left(-\frac{2}{3}\right)$$

Use the distributive property to multiply 3x+15 by 340.

$$1020x+5100=\left(3x-15\right)\times 300+3\left(x-5\right)\left(x+5\right)\left(-\frac{2}{3}\right)$$

Use the distributive property to multiply 3x-15 by 300.

$$1020x+5100=900x-4500+3\left(x-5\right)\left(x+5\right)\left(-\frac{2}{3}\right)$$

Multiply 3 and -\frac{2}{3} to get -2.

$$1020x+5100=900x-4500-2\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply -2 by x-5.

$$1020x+5100=900x-4500+\left(-2x+10\right)\left(x+5\right)$$

Use the distributive property to multiply -2x+10 by x+5 and combine like terms.

$$1020x+5100=900x-4500-2x^{2}+50$$

Add -4500 and 50 to get -4450.

$$1020x+5100=900x-4450-2x^{2}$$

Subtract 900x from both sides.

$$1020x+5100-900x=-4450-2x^{2}$$

Combine 1020x and -900x to get 120x.

$$120x+5100=-4450-2x^{2}$$

Add 2x^{2} to both sides.

$$120x+5100+2x^{2}=-4450$$

Subtract 5100 from both sides.

$$120x+2x^{2}=-4450-5100$$

Subtract 5100 from -4450 to get -9550.

$$120x+2x^{2}=-9550$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$2x^{2}+120x=-9550$$

Divide both sides by 2.

$$\frac{2x^{2}+120x}{2}=-\frac{9550}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{120}{2}x=-\frac{9550}{2}$$

Divide 120 by 2.

$$x^{2}+60x=-\frac{9550}{2}$$

Divide -9550 by 2.

$$x^{2}+60x=-4775$$

Divide 60, the coefficient of the x term, by 2 to get 30. Then add the square of 30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+60x+30^{2}=-4775+30^{2}$$

Square 30.

$$x^{2}+60x+900=-4775+900$$

Add -4775 to 900.

$$x^{2}+60x+900=-3875$$

Factor x^{2}+60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+30\right)^{2}=-3875$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+30\right)^{2}}=\sqrt{-3875}$$

Simplify.

$$x+30=5\sqrt{155}i x+30=-5\sqrt{155}i$$

Subtract 30 from both sides of the equation.

$$x=-30+5\sqrt{155}i x=-5\sqrt{155}i-30$$