Solve n^2-56=-n

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

Solve for n

$$n=-8$$

$$n=7$$

Steps Using Factoring

Add n to both sides.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

To solve the equation, factor n^{2}+n-56 using formula n^{2}+\left(a+b\right)n+ab=\left(n+a\right)\left(n+b\right). To find a and b, set up a system to be solved.

$$a+b=1 ab=-56$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -56.

$$-1,56 -2,28 -4,14 -7,8$$

Calculate the sum for each pair.

$$-1+56=55 -2+28=26 -4+14=10 -7+8=1$$

The solution is the pair that gives sum 1.

$$a=-7 b=8$$

Rewrite factored expression \left(n+a\right)\left(n+b\right) using the obtained values.

$$\left(n-7\right)\left(n+8\right)$$

To find equation solutions, solve n-7=0 and n+8=0.

$$n=7 n=-8$$

Steps Using Factoring By Grouping

Add n to both sides.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as n^{2}+an+bn-56. To find a and b, set up a system to be solved.

$$a+b=1 ab=1\left(-56\right)=-56$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -56.

$$-1,56 -2,28 -4,14 -7,8$$

Calculate the sum for each pair.

$$-1+56=55 -2+28=26 -4+14=10 -7+8=1$$

The solution is the pair that gives sum 1.

$$a=-7 b=8$$

Rewrite n^{2}+n-56 as \left(n^{2}-7n\right)+\left(8n-56\right).

$$\left(n^{2}-7n\right)+\left(8n-56\right)$$

Factor out n in the first and 8 in the second group.

$$n\left(n-7\right)+8\left(n-7\right)$$

Factor out common term n-7 by using distributive property.

$$\left(n-7\right)\left(n+8\right)$$

To find equation solutions, solve n-7=0 and n+8=0.

$$n=7 n=-8$$

Steps Using the Quadratic Formula

Add n to both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-56\right)}}{2}$$

Square 1.

$$n=\frac{-1±\sqrt{1-4\left(-56\right)}}{2}$$

Multiply -4 times -56.

$$n=\frac{-1±\sqrt{1+224}}{2}$$

Add 1 to 224.

$$n=\frac{-1±\sqrt{225}}{2}$$

Take the square root of 225.

$$n=\frac{-1±15}{2}$$

Now solve the equation n=\frac{-1±15}{2} when ± is plus. Add -1 to 15.

$$n=\frac{14}{2}$$

Divide 14 by 2.

$$n=7$$

Now solve the equation n=\frac{-1±15}{2} when ± is minus. Subtract 15 from -1.

$$n=-\frac{16}{2}$$

Divide -16 by 2.

$$n=-8$$

The equation is now solved.

$$n=7 n=-8$$

Steps for Completing the Square

Add n to both sides.

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

Add 56 to both sides. Anything plus zero gives itself.

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

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=56+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+n+\frac{1}{4}=56+\frac{1}{4}$$

Add 56 to \frac{1}{4}.

$$n^{2}+n+\frac{1}{4}=\frac{225}{4}$$

Factor n^{2}+n+\frac{1}{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}.

$$\left(n+\frac{1}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$n+\frac{1}{2}=\frac{15}{2} n+\frac{1}{2}=-\frac{15}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$n=7 n=-8$$