Solve n^2+8=6n

$$n ^ { 2 } + 8 = 6 n$$

Solve for n

$$n=2$$

$$n=4$$

Steps Using Factoring

Subtract 6n from both sides.

$$n^{2}+8-6n=0$$

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

$$n^{2}-6n+8=0$$

To solve the equation, factor n^{2}-6n+8 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.

$$a+b=-6 ab=8$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 8.

$$-1,-8 -2,-4$$

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -6.

$$a=-4 b=-2$$

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

$$\left(n-4\right)\left(n-2\right)$$

To find equation solutions, solve n-4=0 and n-2=0.

$$n=4 n=2$$

Steps Using Factoring By Grouping

Subtract 6n from both sides.

$$n^{2}+8-6n=0$$

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

$$n^{2}-6n+8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as n^{2}+an+bn+8. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\times 8=8$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 8.

$$-1,-8 -2,-4$$

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -6.

$$a=-4 b=-2$$

Rewrite n^{2}-6n+8 as \left(n^{2}-4n\right)+\left(-2n+8\right).

$$\left(n^{2}-4n\right)+\left(-2n+8\right)$$

Factor out n in the first and -2 in the second group.

$$n\left(n-4\right)-2\left(n-4\right)$$

Factor out common term n-4 by using distributive property.

$$\left(n-4\right)\left(n-2\right)$$

To find equation solutions, solve n-4=0 and n-2=0.

$$n=4 n=2$$

Steps Using the Quadratic Formula

Subtract 6n from both sides.

$$n^{2}+8-6n=0$$

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.

$$n^{2}-6n+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 8}}{2}$$

Square -6.

$$n=\frac{-\left(-6\right)±\sqrt{36-4\times 8}}{2}$$

Multiply -4 times 8.

$$n=\frac{-\left(-6\right)±\sqrt{36-32}}{2}$$

Add 36 to -32.

$$n=\frac{-\left(-6\right)±\sqrt{4}}{2}$$

Take the square root of 4.

$$n=\frac{-\left(-6\right)±2}{2}$$

The opposite of -6 is 6.

$$n=\frac{6±2}{2}$$

Now solve the equation n=\frac{6±2}{2} when ± is plus. Add 6 to 2.

$$n=\frac{8}{2}$$

Divide 8 by 2.

$$n=4$$

Now solve the equation n=\frac{6±2}{2} when ± is minus. Subtract 2 from 6.

$$n=\frac{4}{2}$$

Divide 4 by 2.

$$n=2$$

The equation is now solved.

$$n=4 n=2$$

Steps for Completing the Square

Subtract 6n from both sides.

$$n^{2}+8-6n=0$$

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$n^{2}-6n=-8$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-6n+\left(-3\right)^{2}=-8+\left(-3\right)^{2}$$

Square -3.

$$n^{2}-6n+9=-8+9$$

Add -8 to 9.

$$n^{2}-6n+9=1$$

Factor n^{2}-6n+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-3\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-3\right)^{2}}=\sqrt{1}$$

Simplify.

$$n-3=1 n-3=-1$$

Add 3 to both sides of the equation.

$$n=4 n=2$$