Solve 5n-1/n=-5+-1-5n/-5

$$\frac{ 5n-1 }{ n } = -5+ \frac{ -1-5n }{ -5 }$$

Solve for n

$$n = \frac{\sqrt{2301} + 49}{10} \approx 9.696873982$$

$$n=\frac{49-\sqrt{2301}}{10}\approx 0.103126018$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5n, the least common multiple of n,-5.

$$5\left(5n-1\right)=5n\left(-5\right)-n\left(-1-5n\right)$$

Use the distributive property to multiply 5 by 5n-1.

$$25n-5=5n\left(-5\right)-n\left(-1-5n\right)$$

Multiply 5 and -5 to get -25.

$$25n-5=-25n-n\left(-1-5n\right)$$

Use the distributive property to multiply -n by -1-5n.

$$25n-5=-25n+n+5n^{2}$$

Combine -25n and n to get -24n.

$$25n-5=-24n+5n^{2}$$

Add 24n to both sides.

$$25n-5+24n=5n^{2}$$

Combine 25n and 24n to get 49n.

$$49n-5=5n^{2}$$

Subtract 5n^{2} from both sides.

$$49n-5-5n^{2}=0$$

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.

$$-5n^{2}+49n-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 49 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-49±\sqrt{49^{2}-4\left(-5\right)\left(-5\right)}}{2\left(-5\right)}$$

Square 49.

$$n=\frac{-49±\sqrt{2401-4\left(-5\right)\left(-5\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$n=\frac{-49±\sqrt{2401+20\left(-5\right)}}{2\left(-5\right)}$$

Multiply 20 times -5.

$$n=\frac{-49±\sqrt{2401-100}}{2\left(-5\right)}$$

Add 2401 to -100.

$$n=\frac{-49±\sqrt{2301}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$n=\frac{-49±\sqrt{2301}}{-10}$$

Now solve the equation n=\frac{-49±\sqrt{2301}}{-10} when ± is plus. Add -49 to \sqrt{2301}.

$$n=\frac{\sqrt{2301}-49}{-10}$$

Divide -49+\sqrt{2301} by -10.

$$n=\frac{49-\sqrt{2301}}{10}$$

Now solve the equation n=\frac{-49±\sqrt{2301}}{-10} when ± is minus. Subtract \sqrt{2301} from -49.

$$n=\frac{-\sqrt{2301}-49}{-10}$$

Divide -49-\sqrt{2301} by -10.

$$n=\frac{\sqrt{2301}+49}{10}$$

The equation is now solved.

$$n=\frac{49-\sqrt{2301}}{10} n=\frac{\sqrt{2301}+49}{10}$$

Steps for Completing the Square

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5n, the least common multiple of n,-5.

$$5\left(5n-1\right)=5n\left(-5\right)-n\left(-1-5n\right)$$

Use the distributive property to multiply 5 by 5n-1.

$$25n-5=5n\left(-5\right)-n\left(-1-5n\right)$$

Multiply 5 and -5 to get -25.

$$25n-5=-25n-n\left(-1-5n\right)$$

Use the distributive property to multiply -n by -1-5n.

$$25n-5=-25n+n+5n^{2}$$

Combine -25n and n to get -24n.

$$25n-5=-24n+5n^{2}$$

Add 24n to both sides.

$$25n-5+24n=5n^{2}$$

Combine 25n and 24n to get 49n.

$$49n-5=5n^{2}$$

Subtract 5n^{2} from both sides.

$$49n-5-5n^{2}=0$$

Add 5 to both sides. Anything plus zero gives itself.

$$49n-5n^{2}=5$$

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.

$$-5n^{2}+49n=5$$

Divide both sides by -5.

$$\frac{-5n^{2}+49n}{-5}=\frac{5}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$n^{2}+\frac{49}{-5}n=\frac{5}{-5}$$

Divide 49 by -5.

$$n^{2}-\frac{49}{5}n=\frac{5}{-5}$$

Divide 5 by -5.

$$n^{2}-\frac{49}{5}n=-1$$

Divide -\frac{49}{5}, the coefficient of the x term, by 2 to get -\frac{49}{10}. Then add the square of -\frac{49}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{49}{5}n+\left(-\frac{49}{10}\right)^{2}=-1+\left(-\frac{49}{10}\right)^{2}$$

Square -\frac{49}{10} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{49}{5}n+\frac{2401}{100}=-1+\frac{2401}{100}$$

Add -1 to \frac{2401}{100}.

$$n^{2}-\frac{49}{5}n+\frac{2401}{100}=\frac{2301}{100}$$

Factor n^{2}-\frac{49}{5}n+\frac{2401}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{49}{10}\right)^{2}=\frac{2301}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{49}{10}\right)^{2}}=\sqrt{\frac{2301}{100}}$$

Simplify.

$$n-\frac{49}{10}=\frac{\sqrt{2301}}{10} n-\frac{49}{10}=-\frac{\sqrt{2301}}{10}$$

Add \frac{49}{10} to both sides of the equation.

$$n=\frac{\sqrt{2301}+49}{10} n=\frac{49-\sqrt{2301}}{10}$$