Solve -14=n/2[-2n+10)

$$- 14 = \frac { n } { 2 } [ - 2 n + 10 )$$

Solve for n

$$n=-2$$

$$n=7$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$-28=n\left(-2n+10\right)$$

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

$$-28=-2n^{2}+10n$$

Swap sides so that all variable terms are on the left hand side.

$$-2n^{2}+10n=-28$$

Add 28 to both sides.

$$-2n^{2}+10n+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 10 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-10±\sqrt{10^{2}-4\left(-2\right)\times 28}}{2\left(-2\right)}$$

Square 10.

$$n=\frac{-10±\sqrt{100-4\left(-2\right)\times 28}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$n=\frac{-10±\sqrt{100+8\times 28}}{2\left(-2\right)}$$

Multiply 8 times 28.

$$n=\frac{-10±\sqrt{100+224}}{2\left(-2\right)}$$

Add 100 to 224.

$$n=\frac{-10±\sqrt{324}}{2\left(-2\right)}$$

Take the square root of 324.

$$n=\frac{-10±18}{2\left(-2\right)}$$

Multiply 2 times -2.

$$n=\frac{-10±18}{-4}$$

Now solve the equation n=\frac{-10±18}{-4} when ± is plus. Add -10 to 18.

$$n=\frac{8}{-4}$$

Divide 8 by -4.

$$n=-2$$

Now solve the equation n=\frac{-10±18}{-4} when ± is minus. Subtract 18 from -10.

$$n=-\frac{28}{-4}$$

Divide -28 by -4.

$$n=7$$

The equation is now solved.

$$n=-2 n=7$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$-28=n\left(-2n+10\right)$$

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

$$-28=-2n^{2}+10n$$

Swap sides so that all variable terms are on the left hand side.

$$-2n^{2}+10n=-28$$

Divide both sides by -2.

$$\frac{-2n^{2}+10n}{-2}=-\frac{28}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$n^{2}+\frac{10}{-2}n=-\frac{28}{-2}$$

Divide 10 by -2.

$$n^{2}-5n=-\frac{28}{-2}$$

Divide -28 by -2.

$$n^{2}-5n=14$$

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.

$$n^{2}-5n+\left(-\frac{5}{2}\right)^{2}=14+\left(-\frac{5}{2}\right)^{2}$$

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

$$n^{2}-5n+\frac{25}{4}=14+\frac{25}{4}$$

Add 14 to \frac{25}{4}.

$$n^{2}-5n+\frac{25}{4}=\frac{81}{4}$$

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}.

$$\left(n-\frac{5}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{5}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$n-\frac{5}{2}=\frac{9}{2} n-\frac{5}{2}=-\frac{9}{2}$$

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

$$n=7 n=-2$$