Solve n(n-1)-(n-1)*2=56

$$n ( n - 1 ) - ( n - 1 ) \times 2 = 56$$

Solve for n

$$n=-6$$

$$n=9$$

Steps Using the Quadratic Formula

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

$$n^{2}-n-\left(n-1\right)\times 2=56$$

Use the distributive property to multiply n-1 by 2.

$$n^{2}-n-\left(2n-2\right)=56$$

To find the opposite of 2n-2, find the opposite of each term.

$$n^{2}-n-2n-\left(-2\right)=56$$

The opposite of -2 is 2.

$$n^{2}-n-2n+2=56$$

Combine -n and -2n to get -3n.

$$n^{2}-3n+2=56$$

Subtract 56 from both sides.

$$n^{2}-3n+2-56=0$$

Subtract 56 from 2 to get -54.

$$n^{2}-3n-54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-54\right)}}{2}$$

Square -3.

$$n=\frac{-\left(-3\right)±\sqrt{9-4\left(-54\right)}}{2}$$

Multiply -4 times -54.

$$n=\frac{-\left(-3\right)±\sqrt{9+216}}{2}$$

Add 9 to 216.

$$n=\frac{-\left(-3\right)±\sqrt{225}}{2}$$

Take the square root of 225.

$$n=\frac{-\left(-3\right)±15}{2}$$

The opposite of -3 is 3.

$$n=\frac{3±15}{2}$$

Now solve the equation n=\frac{3±15}{2} when ± is plus. Add 3 to 15.

$$n=\frac{18}{2}$$

Divide 18 by 2.

$$n=9$$

Now solve the equation n=\frac{3±15}{2} when ± is minus. Subtract 15 from 3.

$$n=-\frac{12}{2}$$

Divide -12 by 2.

$$n=-6$$

The equation is now solved.

$$n=9 n=-6$$

Steps for Completing the Square

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

$$n^{2}-n-\left(n-1\right)\times 2=56$$

Use the distributive property to multiply n-1 by 2.

$$n^{2}-n-\left(2n-2\right)=56$$

To find the opposite of 2n-2, find the opposite of each term.

$$n^{2}-n-2n-\left(-2\right)=56$$

The opposite of -2 is 2.

$$n^{2}-n-2n+2=56$$

Combine -n and -2n to get -3n.

$$n^{2}-3n+2=56$$

Subtract 2 from both sides.

$$n^{2}-3n=56-2$$

Subtract 2 from 56 to get 54.

$$n^{2}-3n=54$$

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

$$n^{2}-3n+\left(-\frac{3}{2}\right)^{2}=54+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-3n+\frac{9}{4}=54+\frac{9}{4}$$

Add 54 to \frac{9}{4}.

$$n^{2}-3n+\frac{9}{4}=\frac{225}{4}$$

Factor n^{2}-3n+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{3}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{3}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$n-\frac{3}{2}=\frac{15}{2} n-\frac{3}{2}=-\frac{15}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$n=9 n=-6$$