Solve 2*(n+2)*(n+3/3)=64

$$2 \cdot ( n + 2 ) \cdot ( \frac { n + 3 } { 3 } ) = 64$$

Solve for n

$$n = \frac{\sqrt{385} - 5}{2} \approx 7.310708435$$

$$n=\frac{-\sqrt{385}-5}{2}\approx -12.310708435$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$2\left(n+2\right)\left(n+3\right)=192$$

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

$$\left(2n+4\right)\left(n+3\right)=192$$

Apply the distributive property by multiplying each term of 2n+4 by each term of n+3.

$$2n^{2}+6n+4n+12=192$$

Combine 6n and 4n to get 10n.

$$2n^{2}+10n+12=192$$

Subtract 192 from both sides.

$$2n^{2}+10n+12-192=0$$

Subtract 192 from 12 to get -180.

$$2n^{2}+10n-180=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 10 for b, and -180 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-10±\sqrt{10^{2}-4\times 2\left(-180\right)}}{2\times 2}$$

Square 10.

$$n=\frac{-10±\sqrt{100-4\times 2\left(-180\right)}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{-10±\sqrt{100-8\left(-180\right)}}{2\times 2}$$

Multiply -8 times -180.

$$n=\frac{-10±\sqrt{100+1440}}{2\times 2}$$

Add 100 to 1440.

$$n=\frac{-10±\sqrt{1540}}{2\times 2}$$

Take the square root of 1540.

$$n=\frac{-10±2\sqrt{385}}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{-10±2\sqrt{385}}{4}$$

Now solve the equation n=\frac{-10±2\sqrt{385}}{4} when ± is plus. Add -10 to 2\sqrt{385}.

$$n=\frac{2\sqrt{385}-10}{4}$$

Divide -10+2\sqrt{385} by 4.

$$n=\frac{\sqrt{385}-5}{2}$$

Now solve the equation n=\frac{-10±2\sqrt{385}}{4} when ± is minus. Subtract 2\sqrt{385} from -10.

$$n=\frac{-2\sqrt{385}-10}{4}$$

Divide -10-2\sqrt{385} by 4.

$$n=\frac{-\sqrt{385}-5}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{385}-5}{2} n=\frac{-\sqrt{385}-5}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$2\left(n+2\right)\left(n+3\right)=192$$

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

$$\left(2n+4\right)\left(n+3\right)=192$$

Apply the distributive property by multiplying each term of 2n+4 by each term of n+3.

$$2n^{2}+6n+4n+12=192$$

Combine 6n and 4n to get 10n.

$$2n^{2}+10n+12=192$$

Subtract 12 from both sides.

$$2n^{2}+10n=192-12$$

Subtract 12 from 192 to get 180.

$$2n^{2}+10n=180$$

Divide both sides by 2.

$$\frac{2n^{2}+10n}{2}=\frac{180}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$n^{2}+\frac{10}{2}n=\frac{180}{2}$$

Divide 10 by 2.

$$n^{2}+5n=\frac{180}{2}$$

Divide 180 by 2.

$$n^{2}+5n=90$$

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.

$$n^{2}+5n+\left(\frac{5}{2}\right)^{2}=90+\left(\frac{5}{2}\right)^{2}$$

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

$$n^{2}+5n+\frac{25}{4}=90+\frac{25}{4}$$

Add 90 to \frac{25}{4}.

$$n^{2}+5n+\frac{25}{4}=\frac{385}{4}$$

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}.

$$\left(n+\frac{5}{2}\right)^{2}=\frac{385}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{5}{2}\right)^{2}}=\sqrt{\frac{385}{4}}$$

Simplify.

$$n+\frac{5}{2}=\frac{\sqrt{385}}{2} n+\frac{5}{2}=-\frac{\sqrt{385}}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$n=\frac{\sqrt{385}-5}{2} n=\frac{-\sqrt{385}-5}{2}$$