Solve 4n-2=(n+1)(n-2)

$$4 n - 2 = ( n + 1 ) ( n - 2 )$$

Solve for n

$$n=5$$

$$n=0$$

Steps Using the Quadratic Formula

Use the distributive property to multiply n+1 by n-2 and combine like terms.

$$4n-2=n^{2}-n-2$$

Subtract n^{2} from both sides.

$$4n-2-n^{2}=-n-2$$

Add n to both sides.

$$4n-2-n^{2}+n=-2$$

Combine 4n and n to get 5n.

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

Add 2 to both sides.

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

Add -2 and 2 to get 0.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-5±\sqrt{5^{2}}}{2\left(-1\right)}$$

Take the square root of 5^{2}.

$$n=\frac{-5±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-5±5}{-2}$$

Now solve the equation n=\frac{-5±5}{-2} when ± is plus. Add -5 to 5.

$$n=\frac{0}{-2}$$

Divide 0 by -2.

$$n=0$$

Now solve the equation n=\frac{-5±5}{-2} when ± is minus. Subtract 5 from -5.

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

Divide -10 by -2.

$$n=5$$

The equation is now solved.

$$n=0 n=5$$

Steps for Completing the Square

Use the distributive property to multiply n+1 by n-2 and combine like terms.

$$4n-2=n^{2}-n-2$$

Subtract n^{2} from both sides.

$$4n-2-n^{2}=-n-2$$

Add n to both sides.

$$4n-2-n^{2}+n=-2$$

Combine 4n and n to get 5n.

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

Add 2 to both sides.

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

Add -2 and 2 to get 0.

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

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.

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

Divide both sides by -1.

$$\frac{-n^{2}+5n}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\frac{5}{-1}n=\frac{0}{-1}$$

Divide 5 by -1.

$$n^{2}-5n=\frac{0}{-1}$$

Divide 0 by -1.

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

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

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

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-5n+\frac{25}{4}=\frac{25}{4}$$

Factor n^{2}-5n+\frac{25}{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}.

$$\left(n-\frac{5}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{5}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

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

Add \frac{5}{2} to both sides of the equation.

$$n=5 n=0$$