Solve (5n=n^2-n-1)

$$( 5 n = n ^ { 2 } - n - 1 )$$

Solve for n

$$n=\sqrt{10}+3\approx 6.16227766$$

$$n=3-\sqrt{10}\approx -0.16227766$$

Steps Using the Quadratic Formula

Subtract n^{2} from both sides.

$$5n-n^{2}=-n-1$$

Add n to both sides.

$$5n-n^{2}+n=-1$$

Combine 5n and n to get 6n.

$$6n-n^{2}=-1$$

Add 1 to both sides.

$$6n-n^{2}+1=0$$

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.

$$-n^{2}+6n+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-6±\sqrt{6^{2}-4\left(-1\right)}}{2\left(-1\right)}$$

Square 6.

$$n=\frac{-6±\sqrt{36-4\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-6±\sqrt{36+4}}{2\left(-1\right)}$$

Add 36 to 4.

$$n=\frac{-6±\sqrt{40}}{2\left(-1\right)}$$

Take the square root of 40.

$$n=\frac{-6±2\sqrt{10}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-6±2\sqrt{10}}{-2}$$

Now solve the equation n=\frac{-6±2\sqrt{10}}{-2} when ± is plus. Add -6 to 2\sqrt{10}.

$$n=\frac{2\sqrt{10}-6}{-2}$$

Divide -6+2\sqrt{10} by -2.

$$n=3-\sqrt{10}$$

Now solve the equation n=\frac{-6±2\sqrt{10}}{-2} when ± is minus. Subtract 2\sqrt{10} from -6.

$$n=\frac{-2\sqrt{10}-6}{-2}$$

Divide -6-2\sqrt{10} by -2.

$$n=\sqrt{10}+3$$

The equation is now solved.

$$n=3-\sqrt{10} n=\sqrt{10}+3$$

Steps for Completing the Square

Subtract n^{2} from both sides.

$$5n-n^{2}=-n-1$$

Add n to both sides.

$$5n-n^{2}+n=-1$$

Combine 5n and n to get 6n.

$$6n-n^{2}=-1$$

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.

$$-n^{2}+6n=-1$$

Divide both sides by -1.

$$\frac{-n^{2}+6n}{-1}=-\frac{1}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\frac{6}{-1}n=-\frac{1}{-1}$$

Divide 6 by -1.

$$n^{2}-6n=-\frac{1}{-1}$$

Divide -1 by -1.

$$n^{2}-6n=1$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-6n+\left(-3\right)^{2}=1+\left(-3\right)^{2}$$

Square -3.

$$n^{2}-6n+9=1+9$$

Add 1 to 9.

$$n^{2}-6n+9=10$$

Factor n^{2}-6n+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}.

$$\left(n-3\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-3\right)^{2}}=\sqrt{10}$$

Simplify.

$$n-3=\sqrt{10} n-3=-\sqrt{10}$$

Add 3 to both sides of the equation.

$$n=\sqrt{10}+3 n=3-\sqrt{10}$$