Solve 2.4351+1.28825/L-0.959-0.5925L^2+0.959L/frac{L^2{6}}=0

$$\frac{ 2.4351+1.28825 }{ L } - \frac{ 0.959-0.5925 { L }^{ 2 } +0.959L }{ \frac{ { L }^{ 2 } }{ 6 } } = 0$$

Solve for L

$$L = \frac{11 \sqrt{284117089} + 40613}{142200} \approx 1.589497221$$

$$L=\frac{40613-11\sqrt{284117089}}{142200}\approx -1.018287657$$

Steps Using the Quadratic Formula

Variable L cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by L.

$$2.4351+1.28825-6L^{-1}\left(0.959-0.5925L^{2}+0.959L\right)=0$$

Add 2.4351 and 1.28825 to get 3.72335.

$$3.72335-6L^{-1}\left(0.959-0.5925L^{2}+0.959L\right)=0$$

Use the distributive property to multiply 6L^{-1} by 0.959-0.5925L^{2}+0.959L.

$$3.72335-\left(5.754L^{-1}-3.555L+5.754\right)=0$$

To find the opposite of 5.754L^{-1}-3.555L+5.754, find the opposite of each term.

$$3.72335-5.754L^{-1}+3.555L-5.754=0$$

Subtract 5.754 from 3.72335 to get -2.03065.

$$-2.03065-5.754L^{-1}+3.555L=0$$

Reorder the terms.

$$3.555L-2.03065-5.754\times \frac{1}{L}=0$$

Variable L cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by L.

$$3.555LL+L\left(-2.03065\right)-5.754=0$$

Multiply L and L to get L^{2}.

$$3.555L^{2}+L\left(-2.03065\right)-5.754=0$$

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.

$$3.555L^{2}-2.03065L-5.754=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3.555 for a, -2.03065 for b, and -5.754 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$L=\frac{-\left(-2.03065\right)±\sqrt{\left(-2.03065\right)^{2}-4\times 3.555\left(-5.754\right)}}{2\times 3.555}$$

Square -2.03065 by squaring both the numerator and the denominator of the fraction.

$$L=\frac{-\left(-2.03065\right)±\sqrt{4.1235394225-4\times 3.555\left(-5.754\right)}}{2\times 3.555}$$

Multiply -4 times 3.555.

$$L=\frac{-\left(-2.03065\right)±\sqrt{4.1235394225-14.22\left(-5.754\right)}}{2\times 3.555}$$

Multiply -14.22 times -5.754 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$L=\frac{-\left(-2.03065\right)±\sqrt{4.1235394225+81.82188}}{2\times 3.555}$$

Add 4.1235394225 to 81.82188 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$L=\frac{-\left(-2.03065\right)±\sqrt{85.9454194225}}{2\times 3.555}$$

Take the square root of 85.9454194225.

$$L=\frac{-\left(-2.03065\right)±\frac{11\sqrt{284117089}}{20000}}{2\times 3.555}$$

The opposite of -2.03065 is 2.03065.

$$L=\frac{2.03065±\frac{11\sqrt{284117089}}{20000}}{2\times 3.555}$$

Multiply 2 times 3.555.

$$L=\frac{2.03065±\frac{11\sqrt{284117089}}{20000}}{7.11}$$

Now solve the equation L=\frac{2.03065±\frac{11\sqrt{284117089}}{20000}}{7.11} when ± is plus. Add 2.03065 to \frac{11\sqrt{284117089}}{20000}.

$$L=\frac{11\sqrt{284117089}+40613}{7.11\times 20000}$$

Divide \frac{40613+11\sqrt{284117089}}{20000} by 7.11 by multiplying \frac{40613+11\sqrt{284117089}}{20000} by the reciprocal of 7.11.

$$L=\frac{11\sqrt{284117089}+40613}{142200}$$

Now solve the equation L=\frac{2.03065±\frac{11\sqrt{284117089}}{20000}}{7.11} when ± is minus. Subtract \frac{11\sqrt{284117089}}{20000} from 2.03065.

$$L=\frac{40613-11\sqrt{284117089}}{7.11\times 20000}$$

Divide \frac{40613-11\sqrt{284117089}}{20000} by 7.11 by multiplying \frac{40613-11\sqrt{284117089}}{20000} by the reciprocal of 7.11.

$$L=\frac{40613-11\sqrt{284117089}}{142200}$$

The equation is now solved.

$$L=\frac{11\sqrt{284117089}+40613}{142200} L=\frac{40613-11\sqrt{284117089}}{142200}$$

Steps for Completing the Square

Variable L cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by L.

$$2.4351+1.28825-6L^{-1}\left(0.959-0.5925L^{2}+0.959L\right)=0$$

Add 2.4351 and 1.28825 to get 3.72335.

$$3.72335-6L^{-1}\left(0.959-0.5925L^{2}+0.959L\right)=0$$

Use the distributive property to multiply 6L^{-1} by 0.959-0.5925L^{2}+0.959L.

$$3.72335-\left(5.754L^{-1}-3.555L+5.754\right)=0$$

To find the opposite of 5.754L^{-1}-3.555L+5.754, find the opposite of each term.

$$3.72335-5.754L^{-1}+3.555L-5.754=0$$

Subtract 5.754 from 3.72335 to get -2.03065.

$$-2.03065-5.754L^{-1}+3.555L=0$$

Add 2.03065 to both sides. Anything plus zero gives itself.

$$-5.754L^{-1}+3.555L=2.03065$$

Reorder the terms.

$$3.555L-5.754\times \frac{1}{L}=2.03065$$

Variable L cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by L.

$$3.555LL-5.754=2.03065L$$

Multiply L and L to get L^{2}.

$$3.555L^{2}-5.754=2.03065L$$

Subtract 2.03065L from both sides.

$$3.555L^{2}-5.754-2.03065L=0$$

Add 5.754 to both sides. Anything plus zero gives itself.

$$3.555L^{2}-2.03065L=5.754$$

Divide both sides of the equation by 3.555, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{3.555L^{2}-2.03065L}{3.555}=\frac{5.754}{3.555}$$

Dividing by 3.555 undoes the multiplication by 3.555.

$$L^{2}+\left(-\frac{2.03065}{3.555}\right)L=\frac{5.754}{3.555}$$

Divide -2.03065 by 3.555 by multiplying -2.03065 by the reciprocal of 3.555.

$$L^{2}-\frac{40613}{71100}L=\frac{5.754}{3.555}$$

Divide 5.754 by 3.555 by multiplying 5.754 by the reciprocal of 3.555.

$$L^{2}-\frac{40613}{71100}L=\frac{1918}{1185}$$

Divide -\frac{40613}{71100}, the coefficient of the x term, by 2 to get -\frac{40613}{142200}. Then add the square of -\frac{40613}{142200} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$L^{2}-\frac{40613}{71100}L+\left(-\frac{40613}{142200}\right)^{2}=\frac{1918}{1185}+\left(-\frac{40613}{142200}\right)^{2}$$

Square -\frac{40613}{142200} by squaring both the numerator and the denominator of the fraction.

$$L^{2}-\frac{40613}{71100}L+\frac{1649415769}{20220840000}=\frac{1918}{1185}+\frac{1649415769}{20220840000}$$

Add \frac{1918}{1185} to \frac{1649415769}{20220840000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$L^{2}-\frac{40613}{71100}L+\frac{1649415769}{20220840000}=\frac{34378167769}{20220840000}$$

Factor L^{2}-\frac{40613}{71100}L+\frac{1649415769}{20220840000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(L-\frac{40613}{142200}\right)^{2}=\frac{34378167769}{20220840000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(L-\frac{40613}{142200}\right)^{2}}=\sqrt{\frac{34378167769}{20220840000}}$$

Simplify.

$$L-\frac{40613}{142200}=\frac{11\sqrt{284117089}}{142200} L-\frac{40613}{142200}=-\frac{11\sqrt{284117089}}{142200}$$

Add \frac{40613}{142200} to both sides of the equation.

$$L=\frac{11\sqrt{284117089}+40613}{142200} L=\frac{40613-11\sqrt{284117089}}{142200}$$