Solve 3=340^2/340-x-340^2/340+x

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

Solve for x

$$x = \frac{340 \sqrt{115609} - 115600}{3} \approx 1.499970806$$

$$x=\frac{-340\sqrt{115609}-115600}{3}\approx -77068.166637472$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -340,340 since division by zero is not defined. Multiply both sides of the equation by \left(x-340\right)\left(x+340\right), the least common multiple of 340-x,340+x.

$$3\left(x-340\right)\left(x+340\right)=-\left(340+x\right)\times 340^{2}-\left(x-340\right)\times 340^{2}$$

Use the distributive property to multiply 3 by x-340.

$$\left(3x-1020\right)\left(x+340\right)=-\left(340+x\right)\times 340^{2}-\left(x-340\right)\times 340^{2}$$

Use the distributive property to multiply 3x-1020 by x+340 and combine like terms.

$$3x^{2}-346800=-\left(340+x\right)\times 340^{2}-\left(x-340\right)\times 340^{2}$$

Calculate 340 to the power of 2 and get 115600.

$$3x^{2}-346800=-\left(340+x\right)\times 115600-\left(x-340\right)\times 340^{2}$$

Multiply -1 and 115600 to get -115600.

$$3x^{2}-346800=-115600\left(340+x\right)-\left(x-340\right)\times 340^{2}$$

Use the distributive property to multiply -115600 by 340+x.

$$3x^{2}-346800=-39304000-115600x-\left(x-340\right)\times 340^{2}$$

Calculate 340 to the power of 2 and get 115600.

$$3x^{2}-346800=-39304000-115600x-\left(x-340\right)\times 115600$$

Use the distributive property to multiply x-340 by 115600.

$$3x^{2}-346800=-39304000-115600x-\left(115600x-39304000\right)$$

To find the opposite of 115600x-39304000, find the opposite of each term.

$$3x^{2}-346800=-39304000-115600x-115600x+39304000$$

Combine -115600x and -115600x to get -231200x.

$$3x^{2}-346800=-39304000-231200x+39304000$$

Add -39304000 and 39304000 to get 0.

$$3x^{2}-346800=-231200x$$

Add 231200x to both sides.

$$3x^{2}-346800+231200x=0$$

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.

$$3x^{2}+231200x-346800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 231200 for b, and -346800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-231200±\sqrt{231200^{2}-4\times 3\left(-346800\right)}}{2\times 3}$$

Square 231200.

$$x=\frac{-231200±\sqrt{53453440000-4\times 3\left(-346800\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-231200±\sqrt{53453440000-12\left(-346800\right)}}{2\times 3}$$

Multiply -12 times -346800.

$$x=\frac{-231200±\sqrt{53453440000+4161600}}{2\times 3}$$

Add 53453440000 to 4161600.

$$x=\frac{-231200±\sqrt{53457601600}}{2\times 3}$$

Take the square root of 53457601600.

$$x=\frac{-231200±680\sqrt{115609}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-231200±680\sqrt{115609}}{6}$$

Now solve the equation x=\frac{-231200±680\sqrt{115609}}{6} when ± is plus. Add -231200 to 680\sqrt{115609}.

$$x=\frac{680\sqrt{115609}-231200}{6}$$

Divide -231200+680\sqrt{115609} by 6.

$$x=\frac{340\sqrt{115609}-115600}{3}$$

Now solve the equation x=\frac{-231200±680\sqrt{115609}}{6} when ± is minus. Subtract 680\sqrt{115609} from -231200.

$$x=\frac{-680\sqrt{115609}-231200}{6}$$

Divide -231200-680\sqrt{115609} by 6.

$$x=\frac{-340\sqrt{115609}-115600}{3}$$

The equation is now solved.

$$x=\frac{340\sqrt{115609}-115600}{3} x=\frac{-340\sqrt{115609}-115600}{3}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -340,340 since division by zero is not defined. Multiply both sides of the equation by \left(x-340\right)\left(x+340\right), the least common multiple of 340-x,340+x.

$$3\left(x-340\right)\left(x+340\right)=-\left(340+x\right)\times 340^{2}-\left(x-340\right)\times 340^{2}$$

Use the distributive property to multiply 3 by x-340.

$$\left(3x-1020\right)\left(x+340\right)=-\left(340+x\right)\times 340^{2}-\left(x-340\right)\times 340^{2}$$

Use the distributive property to multiply 3x-1020 by x+340 and combine like terms.

$$3x^{2}-346800=-\left(340+x\right)\times 340^{2}-\left(x-340\right)\times 340^{2}$$

Calculate 340 to the power of 2 and get 115600.

$$3x^{2}-346800=-\left(340+x\right)\times 115600-\left(x-340\right)\times 340^{2}$$

Multiply -1 and 115600 to get -115600.

$$3x^{2}-346800=-115600\left(340+x\right)-\left(x-340\right)\times 340^{2}$$

Use the distributive property to multiply -115600 by 340+x.

$$3x^{2}-346800=-39304000-115600x-\left(x-340\right)\times 340^{2}$$

Calculate 340 to the power of 2 and get 115600.

$$3x^{2}-346800=-39304000-115600x-\left(x-340\right)\times 115600$$

Use the distributive property to multiply x-340 by 115600.

$$3x^{2}-346800=-39304000-115600x-\left(115600x-39304000\right)$$

To find the opposite of 115600x-39304000, find the opposite of each term.

$$3x^{2}-346800=-39304000-115600x-115600x+39304000$$

Combine -115600x and -115600x to get -231200x.

$$3x^{2}-346800=-39304000-231200x+39304000$$

Add -39304000 and 39304000 to get 0.

$$3x^{2}-346800=-231200x$$

Add 231200x to both sides.

$$3x^{2}-346800+231200x=0$$

Add 346800 to both sides. Anything plus zero gives itself.

$$3x^{2}+231200x=346800$$

Divide both sides by 3.

$$\frac{3x^{2}+231200x}{3}=\frac{346800}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{231200}{3}x=\frac{346800}{3}$$

Divide 346800 by 3.

$$x^{2}+\frac{231200}{3}x=115600$$

Divide \frac{231200}{3}, the coefficient of the x term, by 2 to get \frac{115600}{3}. Then add the square of \frac{115600}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{231200}{3}x+\left(\frac{115600}{3}\right)^{2}=115600+\left(\frac{115600}{3}\right)^{2}$$

Square \frac{115600}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{231200}{3}x+\frac{13363360000}{9}=115600+\frac{13363360000}{9}$$

Add 115600 to \frac{13363360000}{9}.

$$x^{2}+\frac{231200}{3}x+\frac{13363360000}{9}=\frac{13364400400}{9}$$

Factor x^{2}+\frac{231200}{3}x+\frac{13363360000}{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}.

$$\left(x+\frac{115600}{3}\right)^{2}=\frac{13364400400}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{115600}{3}\right)^{2}}=\sqrt{\frac{13364400400}{9}}$$

Simplify.

$$x+\frac{115600}{3}=\frac{340\sqrt{115609}}{3} x+\frac{115600}{3}=-\frac{340\sqrt{115609}}{3}$$

Subtract \frac{115600}{3} from both sides of the equation.

$$x=\frac{340\sqrt{115609}-115600}{3} x=\frac{-340\sqrt{115609}-115600}{3}$$