Solve (4)(n+2)(n+5)=280

$$( 4 ) ( n + 2 ) ( n + 5 ) = 280$$

Solve for n

$$n=-12$$

$$n=5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4 by n+2.

$$\left(4n+8\right)\left(n+5\right)=280$$

Use the distributive property to multiply 4n+8 by n+5 and combine like terms.

$$4n^{2}+28n+40=280$$

Subtract 280 from both sides.

$$4n^{2}+28n+40-280=0$$

Subtract 280 from 40 to get -240.

$$4n^{2}+28n-240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 28 for b, and -240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-28±\sqrt{28^{2}-4\times 4\left(-240\right)}}{2\times 4}$$

Square 28.

$$n=\frac{-28±\sqrt{784-4\times 4\left(-240\right)}}{2\times 4}$$

Multiply -4 times 4.

$$n=\frac{-28±\sqrt{784-16\left(-240\right)}}{2\times 4}$$

Multiply -16 times -240.

$$n=\frac{-28±\sqrt{784+3840}}{2\times 4}$$

Add 784 to 3840.

$$n=\frac{-28±\sqrt{4624}}{2\times 4}$$

Take the square root of 4624.

$$n=\frac{-28±68}{2\times 4}$$

Multiply 2 times 4.

$$n=\frac{-28±68}{8}$$

Now solve the equation n=\frac{-28±68}{8} when ± is plus. Add -28 to 68.

$$n=\frac{40}{8}$$

Divide 40 by 8.

$$n=5$$

Now solve the equation n=\frac{-28±68}{8} when ± is minus. Subtract 68 from -28.

$$n=-\frac{96}{8}$$

Divide -96 by 8.

$$n=-12$$

The equation is now solved.

$$n=5 n=-12$$

Steps for Completing the Square

Use the distributive property to multiply 4 by n+2.

$$\left(4n+8\right)\left(n+5\right)=280$$

Use the distributive property to multiply 4n+8 by n+5 and combine like terms.

$$4n^{2}+28n+40=280$$

Subtract 40 from both sides.

$$4n^{2}+28n=280-40$$

Subtract 40 from 280 to get 240.

$$4n^{2}+28n=240$$

Divide both sides by 4.

$$\frac{4n^{2}+28n}{4}=\frac{240}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$n^{2}+\frac{28}{4}n=\frac{240}{4}$$

Divide 28 by 4.

$$n^{2}+7n=\frac{240}{4}$$

Divide 240 by 4.

$$n^{2}+7n=60$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+7n+\left(\frac{7}{2}\right)^{2}=60+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+7n+\frac{49}{4}=60+\frac{49}{4}$$

Add 60 to \frac{49}{4}.

$$n^{2}+7n+\frac{49}{4}=\frac{289}{4}$$

Factor n^{2}+7n+\frac{49}{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}.

$$\left(n+\frac{7}{2}\right)^{2}=\frac{289}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{7}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Simplify.

$$n+\frac{7}{2}=\frac{17}{2} n+\frac{7}{2}=-\frac{17}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$n=5 n=-12$$