Solve 4.8/14n-2+20.8/14n+2=0.3

$$\frac{ 4.8 }{ 14n-2 } + \frac{ 20.8 }{ 14n+2 } = 0.3$$

Solve for n

$$n=\frac{64-\sqrt{3865}}{21}\approx 0.087184563$$

$$n = \frac{\sqrt{3865} + 64}{21} \approx 6.008053532$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values -\frac{1}{7},\frac{1}{7} since division by zero is not defined. Multiply both sides of the equation by 2\left(7n-1\right)\left(7n+1\right), the least common multiple of 14n-2,14n+2.

$$\left(7n+1\right)\times 4.8+\left(7n-1\right)\times 20.8=0.6\left(7n-1\right)\left(7n+1\right)$$

Use the distributive property to multiply 7n+1 by 4.8.

$$33.6n+4.8+\left(7n-1\right)\times 20.8=0.6\left(7n-1\right)\left(7n+1\right)$$

Use the distributive property to multiply 7n-1 by 20.8.

$$33.6n+4.8+145.6n-20.8=0.6\left(7n-1\right)\left(7n+1\right)$$

Combine 33.6n and 145.6n to get 179.2n.

$$179.2n+4.8-20.8=0.6\left(7n-1\right)\left(7n+1\right)$$

Subtract 20.8 from 4.8 to get -16.

$$179.2n-16=0.6\left(7n-1\right)\left(7n+1\right)$$

Use the distributive property to multiply 0.6 by 7n-1.

$$179.2n-16=\left(4.2n-0.6\right)\left(7n+1\right)$$

Use the distributive property to multiply 4.2n-0.6 by 7n+1 and combine like terms.

$$179.2n-16=29.4n^{2}-0.6$$

Subtract 29.4n^{2} from both sides.

$$179.2n-16-29.4n^{2}=-0.6$$

Add 0.6 to both sides.

$$179.2n-16-29.4n^{2}+0.6=0$$

Add -16 and 0.6 to get -15.4.

$$179.2n-15.4-29.4n^{2}=0$$

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.

$$-29.4n^{2}+179.2n-15.4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -29.4 for a, 179.2 for b, and -15.4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-179.2±\sqrt{179.2^{2}-4\left(-29.4\right)\left(-15.4\right)}}{2\left(-29.4\right)}$$

Square 179.2 by squaring both the numerator and the denominator of the fraction.

$$n=\frac{-179.2±\sqrt{32112.64-4\left(-29.4\right)\left(-15.4\right)}}{2\left(-29.4\right)}$$

Multiply -4 times -29.4.

$$n=\frac{-179.2±\sqrt{32112.64+117.6\left(-15.4\right)}}{2\left(-29.4\right)}$$

Multiply 117.6 times -15.4 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$n=\frac{-179.2±\sqrt{\frac{802816-45276}{25}}}{2\left(-29.4\right)}$$

Add 32112.64 to -1811.04 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n=\frac{-179.2±\sqrt{30301.6}}{2\left(-29.4\right)}$$

Take the square root of 30301.6.

$$n=\frac{-179.2±\frac{14\sqrt{3865}}{5}}{2\left(-29.4\right)}$$

Multiply 2 times -29.4.

$$n=\frac{-179.2±\frac{14\sqrt{3865}}{5}}{-58.8}$$

Now solve the equation n=\frac{-179.2±\frac{14\sqrt{3865}}{5}}{-58.8} when ± is plus. Add -179.2 to \frac{14\sqrt{3865}}{5}.

$$n=\frac{14\sqrt{3865}-896}{-58.8\times 5}$$

Divide \frac{-896+14\sqrt{3865}}{5} by -58.8 by multiplying \frac{-896+14\sqrt{3865}}{5} by the reciprocal of -58.8.

$$n=\frac{64-\sqrt{3865}}{21}$$

Now solve the equation n=\frac{-179.2±\frac{14\sqrt{3865}}{5}}{-58.8} when ± is minus. Subtract \frac{14\sqrt{3865}}{5} from -179.2.

$$n=\frac{-14\sqrt{3865}-896}{-58.8\times 5}$$

Divide \frac{-896-14\sqrt{3865}}{5} by -58.8 by multiplying \frac{-896-14\sqrt{3865}}{5} by the reciprocal of -58.8.

$$n=\frac{\sqrt{3865}+64}{21}$$

The equation is now solved.

$$n=\frac{64-\sqrt{3865}}{21} n=\frac{\sqrt{3865}+64}{21}$$

Steps for Completing the Square

Variable n cannot be equal to any of the values -\frac{1}{7},\frac{1}{7} since division by zero is not defined. Multiply both sides of the equation by 2\left(7n-1\right)\left(7n+1\right), the least common multiple of 14n-2,14n+2.

$$\left(7n+1\right)\times 4.8+\left(7n-1\right)\times 20.8=0.6\left(7n-1\right)\left(7n+1\right)$$

Use the distributive property to multiply 7n+1 by 4.8.

$$33.6n+4.8+\left(7n-1\right)\times 20.8=0.6\left(7n-1\right)\left(7n+1\right)$$

Use the distributive property to multiply 7n-1 by 20.8.

$$33.6n+4.8+145.6n-20.8=0.6\left(7n-1\right)\left(7n+1\right)$$

Combine 33.6n and 145.6n to get 179.2n.

$$179.2n+4.8-20.8=0.6\left(7n-1\right)\left(7n+1\right)$$

Subtract 20.8 from 4.8 to get -16.

$$179.2n-16=0.6\left(7n-1\right)\left(7n+1\right)$$

Use the distributive property to multiply 0.6 by 7n-1.

$$179.2n-16=\left(4.2n-0.6\right)\left(7n+1\right)$$

Use the distributive property to multiply 4.2n-0.6 by 7n+1 and combine like terms.

$$179.2n-16=29.4n^{2}-0.6$$

Subtract 29.4n^{2} from both sides.

$$179.2n-16-29.4n^{2}=-0.6$$

Add 16 to both sides.

$$179.2n-29.4n^{2}=-0.6+16$$

Add -0.6 and 16 to get 15.4.

$$179.2n-29.4n^{2}=15.4$$

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.

$$-29.4n^{2}+179.2n=15.4$$

Divide both sides of the equation by -29.4, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-29.4n^{2}+179.2n}{-29.4}=\frac{15.4}{-29.4}$$

Dividing by -29.4 undoes the multiplication by -29.4.

$$n^{2}+\frac{179.2}{-29.4}n=\frac{15.4}{-29.4}$$

Divide 179.2 by -29.4 by multiplying 179.2 by the reciprocal of -29.4.

$$n^{2}-\frac{128}{21}n=\frac{15.4}{-29.4}$$

Divide 15.4 by -29.4 by multiplying 15.4 by the reciprocal of -29.4.

$$n^{2}-\frac{128}{21}n=-\frac{11}{21}$$

Divide -\frac{128}{21}, the coefficient of the x term, by 2 to get -\frac{64}{21}. Then add the square of -\frac{64}{21} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{128}{21}n+\left(-\frac{64}{21}\right)^{2}=-\frac{11}{21}+\left(-\frac{64}{21}\right)^{2}$$

Square -\frac{64}{21} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{128}{21}n+\frac{4096}{441}=-\frac{11}{21}+\frac{4096}{441}$$

Add -\frac{11}{21} to \frac{4096}{441} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{128}{21}n+\frac{4096}{441}=\frac{3865}{441}$$

Factor n^{2}-\frac{128}{21}n+\frac{4096}{441}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{64}{21}\right)^{2}=\frac{3865}{441}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{64}{21}\right)^{2}}=\sqrt{\frac{3865}{441}}$$

Simplify.

$$n-\frac{64}{21}=\frac{\sqrt{3865}}{21} n-\frac{64}{21}=-\frac{\sqrt{3865}}{21}$$

Add \frac{64}{21} to both sides of the equation.

$$n=\frac{\sqrt{3865}+64}{21} n=\frac{64-\sqrt{3865}}{21}$$