Solve 0=3N_0^2-18N_{0}-3240

$$0 = 3 N _ { 0 } ^ { 2 } - 18 N _ { 0 } - 3240$$

Solve for N_0

$$N_{0}=-30$$

$$N_{0}=36$$

Steps Using Factoring By Grouping

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

$$3N_{0}^{2}-18N_{0}-3240=0$$

Divide both sides by 3.

$$N_{0}^{2}-6N_{0}-1080=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as N_{0}^{2}+aN_{0}+bN_{0}-1080. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-1080\right)=-1080$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1080.

$$1,-1080 2,-540 3,-360 4,-270 5,-216 6,-180 8,-135 9,-120 10,-108 12,-90 15,-72 18,-60 20,-54 24,-45 27,-40 30,-36$$

Calculate the sum for each pair.

$$1-1080=-1079 2-540=-538 3-360=-357 4-270=-266 5-216=-211 6-180=-174 8-135=-127 9-120=-111 10-108=-98 12-90=-78 15-72=-57 18-60=-42 20-54=-34 24-45=-21 27-40=-13 30-36=-6$$

The solution is the pair that gives sum -6.

$$a=-36 b=30$$

Rewrite N_{0}^{2}-6N_{0}-1080 as \left(N_{0}^{2}-36N_{0}\right)+\left(30N_{0}-1080\right).

$$\left(N_{0}^{2}-36N_{0}\right)+\left(30N_{0}-1080\right)$$

Factor out N_{0} in the first and 30 in the second group.

$$N_{0}\left(N_{0}-36\right)+30\left(N_{0}-36\right)$$

Factor out common term N_{0}-36 by using distributive property.

$$\left(N_{0}-36\right)\left(N_{0}+30\right)$$

To find equation solutions, solve N_{0}-36=0 and N_{0}+30=0.

$$N_{0}=36 N_{0}=-30$$

Steps Using the Quadratic Formula

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

$$3N_{0}^{2}-18N_{0}-3240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -18 for b, and -3240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$N_{0}=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 3\left(-3240\right)}}{2\times 3}$$

Square -18.

$$N_{0}=\frac{-\left(-18\right)±\sqrt{324-4\times 3\left(-3240\right)}}{2\times 3}$$

Multiply -4 times 3.

$$N_{0}=\frac{-\left(-18\right)±\sqrt{324-12\left(-3240\right)}}{2\times 3}$$

Multiply -12 times -3240.

$$N_{0}=\frac{-\left(-18\right)±\sqrt{324+38880}}{2\times 3}$$

Add 324 to 38880.

$$N_{0}=\frac{-\left(-18\right)±\sqrt{39204}}{2\times 3}$$

Take the square root of 39204.

$$N_{0}=\frac{-\left(-18\right)±198}{2\times 3}$$

The opposite of -18 is 18.

$$N_{0}=\frac{18±198}{2\times 3}$$

Multiply 2 times 3.

$$N_{0}=\frac{18±198}{6}$$

Now solve the equation N_{0}=\frac{18±198}{6} when ± is plus. Add 18 to 198.

$$N_{0}=\frac{216}{6}$$

Divide 216 by 6.

$$N_{0}=36$$

Now solve the equation N_{0}=\frac{18±198}{6} when ± is minus. Subtract 198 from 18.

$$N_{0}=-\frac{180}{6}$$

Divide -180 by 6.

$$N_{0}=-30$$

The equation is now solved.

$$N_{0}=36 N_{0}=-30$$

Steps for Completing the Square

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

$$3N_{0}^{2}-18N_{0}-3240=0$$

Add 3240 to both sides. Anything plus zero gives itself.

$$3N_{0}^{2}-18N_{0}=3240$$

Divide both sides by 3.

$$\frac{3N_{0}^{2}-18N_{0}}{3}=\frac{3240}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$N_{0}^{2}+\left(-\frac{18}{3}\right)N_{0}=\frac{3240}{3}$$

Divide -18 by 3.

$$N_{0}^{2}-6N_{0}=\frac{3240}{3}$$

Divide 3240 by 3.

$$N_{0}^{2}-6N_{0}=1080$$

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.

$$N_{0}^{2}-6N_{0}+\left(-3\right)^{2}=1080+\left(-3\right)^{2}$$

Square -3.

$$N_{0}^{2}-6N_{0}+9=1080+9$$

Add 1080 to 9.

$$N_{0}^{2}-6N_{0}+9=1089$$

Factor N_{0}^{2}-6N_{0}+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}.

$$\left(N_{0}-3\right)^{2}=1089$$

Take the square root of both sides of the equation.

$$\sqrt{\left(N_{0}-3\right)^{2}}=\sqrt{1089}$$

Simplify.

$$N_{0}-3=33 N_{0}-3=-33$$

Add 3 to both sides of the equation.

$$N_{0}=36 N_{0}=-30$$