Solve -n=n(2n-10)

$$- n = n ( 2 n - 10 )$$

Solve for n

$$n = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

$$n=0$$

Steps Using Factoring

Use the distributive property to multiply n by 2n-10.

$$-n=2n^{2}-10n$$

Subtract 2n^{2} from both sides.

$$-n-2n^{2}=-10n$$

Add 10n to both sides.

$$-n-2n^{2}+10n=0$$

Combine -n and 10n to get 9n.

$$9n-2n^{2}=0$$

Factor out n.

$$n\left(9-2n\right)=0$$

To find equation solutions, solve n=0 and 9-2n=0.

$$n=0 n=\frac{9}{2}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply n by 2n-10.

$$-n=2n^{2}-10n$$

Subtract 2n^{2} from both sides.

$$-n-2n^{2}=-10n$$

Add 10n to both sides.

$$-n-2n^{2}+10n=0$$

Combine -n and 10n to get 9n.

$$9n-2n^{2}=0$$

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.

$$-2n^{2}+9n=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 9 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Take the square root of 9^{2}.

$$n=\frac{-9±9}{2\left(-2\right)}$$

Multiply 2 times -2.

$$n=\frac{-9±9}{-4}$$

Now solve the equation n=\frac{-9±9}{-4} when ± is plus. Add -9 to 9.

$$n=\frac{0}{-4}$$

Divide 0 by -4.

$$n=0$$

Now solve the equation n=\frac{-9±9}{-4} when ± is minus. Subtract 9 from -9.

$$n=-\frac{18}{-4}$$

Reduce the fraction \frac{-18}{-4} to lowest terms by extracting and canceling out 2.

$$n=\frac{9}{2}$$

The equation is now solved.

$$n=0 n=\frac{9}{2}$$

Steps for Completing the Square

Use the distributive property to multiply n by 2n-10.

$$-n=2n^{2}-10n$$

Subtract 2n^{2} from both sides.

$$-n-2n^{2}=-10n$$

Add 10n to both sides.

$$-n-2n^{2}+10n=0$$

Combine -n and 10n to get 9n.

$$9n-2n^{2}=0$$

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.

$$-2n^{2}+9n=0$$

Divide both sides by -2.

$$\frac{-2n^{2}+9n}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$n^{2}+\frac{9}{-2}n=\frac{0}{-2}$$

Divide 9 by -2.

$$n^{2}-\frac{9}{2}n=\frac{0}{-2}$$

Divide 0 by -2.

$$n^{2}-\frac{9}{2}n=0$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{9}{2}n+\left(-\frac{9}{4}\right)^{2}=\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{9}{2}n+\frac{81}{16}=\frac{81}{16}$$

Factor n^{2}-\frac{9}{2}n+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{9}{4}\right)^{2}=\frac{81}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{9}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Simplify.

$$n-\frac{9}{4}=\frac{9}{4} n-\frac{9}{4}=-\frac{9}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$n=\frac{9}{2} n=0$$