Solve 3n/n-1-1/3=-40/n-18

$$\frac { 3 n } { n - 1 } - \frac { 1 } { 3 } = \frac { - 40 } { n - 18 }$$

Solve for n

$$n = \frac{\sqrt{4945} + 23}{16} \approx 5.832543373$$

$$n=\frac{23-\sqrt{4945}}{16}\approx -2.957543373$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values 1,18 since division by zero is not defined. Multiply both sides of the equation by 3\left(n-18\right)\left(n-1\right), the least common multiple of n-1,3,n-18.

$$\left(3n-54\right)\times 3n+3\left(n-18\right)\left(n-1\right)\left(-\frac{1}{3}\right)=\left(3n-3\right)\left(-40\right)$$

Use the distributive property to multiply 3n-54 by 3.

$$\left(9n-162\right)n+3\left(n-18\right)\left(n-1\right)\left(-\frac{1}{3}\right)=\left(3n-3\right)\left(-40\right)$$

Use the distributive property to multiply 9n-162 by n.

$$9n^{2}-162n+3\left(n-18\right)\left(n-1\right)\left(-\frac{1}{3}\right)=\left(3n-3\right)\left(-40\right)$$

Multiply 3 and -\frac{1}{3} to get -1.

$$9n^{2}-162n-\left(n-18\right)\left(n-1\right)=\left(3n-3\right)\left(-40\right)$$

Use the distributive property to multiply -1 by n-18.

$$9n^{2}-162n+\left(-n+18\right)\left(n-1\right)=\left(3n-3\right)\left(-40\right)$$

Use the distributive property to multiply -n+18 by n-1 and combine like terms.

$$9n^{2}-162n-n^{2}+19n-18=\left(3n-3\right)\left(-40\right)$$

Combine 9n^{2} and -n^{2} to get 8n^{2}.

$$8n^{2}-162n+19n-18=\left(3n-3\right)\left(-40\right)$$

Combine -162n and 19n to get -143n.

$$8n^{2}-143n-18=\left(3n-3\right)\left(-40\right)$$

Use the distributive property to multiply 3n-3 by -40.

$$8n^{2}-143n-18=-120n+120$$

Add 120n to both sides.

$$8n^{2}-143n-18+120n=120$$

Combine -143n and 120n to get -23n.

$$8n^{2}-23n-18=120$$

Subtract 120 from both sides.

$$8n^{2}-23n-18-120=0$$

Subtract 120 from -18 to get -138.

$$8n^{2}-23n-138=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -23 for b, and -138 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 8\left(-138\right)}}{2\times 8}$$

Square -23.

$$n=\frac{-\left(-23\right)±\sqrt{529-4\times 8\left(-138\right)}}{2\times 8}$$

Multiply -4 times 8.

$$n=\frac{-\left(-23\right)±\sqrt{529-32\left(-138\right)}}{2\times 8}$$

Multiply -32 times -138.

$$n=\frac{-\left(-23\right)±\sqrt{529+4416}}{2\times 8}$$

Add 529 to 4416.

$$n=\frac{-\left(-23\right)±\sqrt{4945}}{2\times 8}$$

The opposite of -23 is 23.

$$n=\frac{23±\sqrt{4945}}{2\times 8}$$

Multiply 2 times 8.

$$n=\frac{23±\sqrt{4945}}{16}$$

Now solve the equation n=\frac{23±\sqrt{4945}}{16} when ± is plus. Add 23 to \sqrt{4945}.

$$n=\frac{\sqrt{4945}+23}{16}$$

Now solve the equation n=\frac{23±\sqrt{4945}}{16} when ± is minus. Subtract \sqrt{4945} from 23.

$$n=\frac{23-\sqrt{4945}}{16}$$

The equation is now solved.

$$n=\frac{\sqrt{4945}+23}{16} n=\frac{23-\sqrt{4945}}{16}$$

Steps for Completing the Square

Variable n cannot be equal to any of the values 1,18 since division by zero is not defined. Multiply both sides of the equation by 3\left(n-18\right)\left(n-1\right), the least common multiple of n-1,3,n-18.

$$\left(3n-54\right)\times 3n+3\left(n-18\right)\left(n-1\right)\left(-\frac{1}{3}\right)=\left(3n-3\right)\left(-40\right)$$

Use the distributive property to multiply 3n-54 by 3.

$$\left(9n-162\right)n+3\left(n-18\right)\left(n-1\right)\left(-\frac{1}{3}\right)=\left(3n-3\right)\left(-40\right)$$

Use the distributive property to multiply 9n-162 by n.

$$9n^{2}-162n+3\left(n-18\right)\left(n-1\right)\left(-\frac{1}{3}\right)=\left(3n-3\right)\left(-40\right)$$

Multiply 3 and -\frac{1}{3} to get -1.

$$9n^{2}-162n-\left(n-18\right)\left(n-1\right)=\left(3n-3\right)\left(-40\right)$$

Use the distributive property to multiply -1 by n-18.

$$9n^{2}-162n+\left(-n+18\right)\left(n-1\right)=\left(3n-3\right)\left(-40\right)$$

Use the distributive property to multiply -n+18 by n-1 and combine like terms.

$$9n^{2}-162n-n^{2}+19n-18=\left(3n-3\right)\left(-40\right)$$

Combine 9n^{2} and -n^{2} to get 8n^{2}.

$$8n^{2}-162n+19n-18=\left(3n-3\right)\left(-40\right)$$

Combine -162n and 19n to get -143n.

$$8n^{2}-143n-18=\left(3n-3\right)\left(-40\right)$$

Use the distributive property to multiply 3n-3 by -40.

$$8n^{2}-143n-18=-120n+120$$

Add 120n to both sides.

$$8n^{2}-143n-18+120n=120$$

Combine -143n and 120n to get -23n.

$$8n^{2}-23n-18=120$$

Add 18 to both sides.

$$8n^{2}-23n=120+18$$

Add 120 and 18 to get 138.

$$8n^{2}-23n=138$$

Divide both sides by 8.

$$\frac{8n^{2}-23n}{8}=\frac{138}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$n^{2}-\frac{23}{8}n=\frac{138}{8}$$

Reduce the fraction \frac{138}{8} to lowest terms by extracting and canceling out 2.

$$n^{2}-\frac{23}{8}n=\frac{69}{4}$$

Divide -\frac{23}{8}, the coefficient of the x term, by 2 to get -\frac{23}{16}. Then add the square of -\frac{23}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{23}{8}n+\left(-\frac{23}{16}\right)^{2}=\frac{69}{4}+\left(-\frac{23}{16}\right)^{2}$$

Square -\frac{23}{16} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{23}{8}n+\frac{529}{256}=\frac{69}{4}+\frac{529}{256}$$

Add \frac{69}{4} to \frac{529}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{23}{8}n+\frac{529}{256}=\frac{4945}{256}$$

Factor n^{2}-\frac{23}{8}n+\frac{529}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{23}{16}\right)^{2}=\frac{4945}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{23}{16}\right)^{2}}=\sqrt{\frac{4945}{256}}$$

Simplify.

$$n-\frac{23}{16}=\frac{\sqrt{4945}}{16} n-\frac{23}{16}=-\frac{\sqrt{4945}}{16}$$

Add \frac{23}{16} to both sides of the equation.

$$n=\frac{\sqrt{4945}+23}{16} n=\frac{23-\sqrt{4945}}{16}$$