Solve 3=340^2/390-x-340^2/300+x

$$3 = \frac { 340 ^ { 2 } } { 390 - x } - \frac { 340 ^ { 2 } } { 300 + x }$$

Solve for x

$$x = \frac{5 \sqrt{534577249} - 115465}{3} \approx 46.544411092$$

$$x=\frac{-5\sqrt{534577249}-115465}{3}\approx -77023.211077758$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -300,390 since division by zero is not defined. Multiply both sides of the equation by \left(x-390\right)\left(x+300\right), the least common multiple of 390-x,300+x.

$$3\left(x-390\right)\left(x+300\right)=-\left(300+x\right)\times 340^{2}-\left(x-390\right)\times 340^{2}$$

Use the distributive property to multiply 3 by x-390.

$$\left(3x-1170\right)\left(x+300\right)=-\left(300+x\right)\times 340^{2}-\left(x-390\right)\times 340^{2}$$

Use the distributive property to multiply 3x-1170 by x+300 and combine like terms.

$$3x^{2}-270x-351000=-\left(300+x\right)\times 340^{2}-\left(x-390\right)\times 340^{2}$$

Calculate 340 to the power of 2 and get 115600.

$$3x^{2}-270x-351000=-\left(300+x\right)\times 115600-\left(x-390\right)\times 340^{2}$$

Multiply -1 and 115600 to get -115600.

$$3x^{2}-270x-351000=-115600\left(300+x\right)-\left(x-390\right)\times 340^{2}$$

Use the distributive property to multiply -115600 by 300+x.

$$3x^{2}-270x-351000=-34680000-115600x-\left(x-390\right)\times 340^{2}$$

Calculate 340 to the power of 2 and get 115600.

$$3x^{2}-270x-351000=-34680000-115600x-\left(x-390\right)\times 115600$$

Use the distributive property to multiply x-390 by 115600.

$$3x^{2}-270x-351000=-34680000-115600x-\left(115600x-45084000\right)$$

To find the opposite of 115600x-45084000, find the opposite of each term.

$$3x^{2}-270x-351000=-34680000-115600x-115600x+45084000$$

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

$$3x^{2}-270x-351000=-34680000-231200x+45084000$$

Add -34680000 and 45084000 to get 10404000.

$$3x^{2}-270x-351000=10404000-231200x$$

Subtract 10404000 from both sides.

$$3x^{2}-270x-351000-10404000=-231200x$$

Subtract 10404000 from -351000 to get -10755000.

$$3x^{2}-270x-10755000=-231200x$$

Add 231200x to both sides.

$$3x^{2}-270x-10755000+231200x=0$$

Combine -270x and 231200x to get 230930x.

$$3x^{2}+230930x-10755000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 230930 for b, and -10755000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-230930±\sqrt{230930^{2}-4\times 3\left(-10755000\right)}}{2\times 3}$$

Square 230930.

$$x=\frac{-230930±\sqrt{53328664900-4\times 3\left(-10755000\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-230930±\sqrt{53328664900-12\left(-10755000\right)}}{2\times 3}$$

Multiply -12 times -10755000.

$$x=\frac{-230930±\sqrt{53328664900+129060000}}{2\times 3}$$

Add 53328664900 to 129060000.

$$x=\frac{-230930±\sqrt{53457724900}}{2\times 3}$$

Take the square root of 53457724900.

$$x=\frac{-230930±10\sqrt{534577249}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-230930±10\sqrt{534577249}}{6}$$

Now solve the equation x=\frac{-230930±10\sqrt{534577249}}{6} when ± is plus. Add -230930 to 10\sqrt{534577249}.

$$x=\frac{10\sqrt{534577249}-230930}{6}$$

Divide -230930+10\sqrt{534577249} by 6.

$$x=\frac{5\sqrt{534577249}-115465}{3}$$

Now solve the equation x=\frac{-230930±10\sqrt{534577249}}{6} when ± is minus. Subtract 10\sqrt{534577249} from -230930.

$$x=\frac{-10\sqrt{534577249}-230930}{6}$$

Divide -230930-10\sqrt{534577249} by 6.

$$x=\frac{-5\sqrt{534577249}-115465}{3}$$

The equation is now solved.

$$x=\frac{5\sqrt{534577249}-115465}{3} x=\frac{-5\sqrt{534577249}-115465}{3}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -300,390 since division by zero is not defined. Multiply both sides of the equation by \left(x-390\right)\left(x+300\right), the least common multiple of 390-x,300+x.

$$3\left(x-390\right)\left(x+300\right)=-\left(300+x\right)\times 340^{2}-\left(x-390\right)\times 340^{2}$$

Use the distributive property to multiply 3 by x-390.

$$\left(3x-1170\right)\left(x+300\right)=-\left(300+x\right)\times 340^{2}-\left(x-390\right)\times 340^{2}$$

Use the distributive property to multiply 3x-1170 by x+300 and combine like terms.

$$3x^{2}-270x-351000=-\left(300+x\right)\times 340^{2}-\left(x-390\right)\times 340^{2}$$

Calculate 340 to the power of 2 and get 115600.

$$3x^{2}-270x-351000=-\left(300+x\right)\times 115600-\left(x-390\right)\times 340^{2}$$

Multiply -1 and 115600 to get -115600.

$$3x^{2}-270x-351000=-115600\left(300+x\right)-\left(x-390\right)\times 340^{2}$$

Use the distributive property to multiply -115600 by 300+x.

$$3x^{2}-270x-351000=-34680000-115600x-\left(x-390\right)\times 340^{2}$$

Calculate 340 to the power of 2 and get 115600.

$$3x^{2}-270x-351000=-34680000-115600x-\left(x-390\right)\times 115600$$

Use the distributive property to multiply x-390 by 115600.

$$3x^{2}-270x-351000=-34680000-115600x-\left(115600x-45084000\right)$$

To find the opposite of 115600x-45084000, find the opposite of each term.

$$3x^{2}-270x-351000=-34680000-115600x-115600x+45084000$$

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

$$3x^{2}-270x-351000=-34680000-231200x+45084000$$

Add -34680000 and 45084000 to get 10404000.

$$3x^{2}-270x-351000=10404000-231200x$$

Add 231200x to both sides.

$$3x^{2}-270x-351000+231200x=10404000$$

Combine -270x and 231200x to get 230930x.

$$3x^{2}+230930x-351000=10404000$$

Add 351000 to both sides.

$$3x^{2}+230930x=10404000+351000$$

Add 10404000 and 351000 to get 10755000.

$$3x^{2}+230930x=10755000$$

Divide both sides by 3.

$$\frac{3x^{2}+230930x}{3}=\frac{10755000}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{230930}{3}x=\frac{10755000}{3}$$

Divide 10755000 by 3.

$$x^{2}+\frac{230930}{3}x=3585000$$

Divide \frac{230930}{3}, the coefficient of the x term, by 2 to get \frac{115465}{3}. Then add the square of \frac{115465}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{230930}{3}x+\left(\frac{115465}{3}\right)^{2}=3585000+\left(\frac{115465}{3}\right)^{2}$$

Square \frac{115465}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{230930}{3}x+\frac{13332166225}{9}=3585000+\frac{13332166225}{9}$$

Add 3585000 to \frac{13332166225}{9}.

$$x^{2}+\frac{230930}{3}x+\frac{13332166225}{9}=\frac{13364431225}{9}$$

Factor x^{2}+\frac{230930}{3}x+\frac{13332166225}{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}.

$$\left(x+\frac{115465}{3}\right)^{2}=\frac{13364431225}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{115465}{3}\right)^{2}}=\sqrt{\frac{13364431225}{9}}$$

Simplify.

$$x+\frac{115465}{3}=\frac{5\sqrt{534577249}}{3} x+\frac{115465}{3}=-\frac{5\sqrt{534577249}}{3}$$

Subtract \frac{115465}{3} from both sides of the equation.

$$x=\frac{5\sqrt{534577249}-115465}{3} x=\frac{-5\sqrt{534577249}-115465}{3}$$