Solve n(-48+(n-1)(4))/2=336

$$\frac { n ( - 48 + ( n - 1 ) ( 4 ) ) } { 2 } = 336$$

Solve for n

$$n=-8$$

$$n=21$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$n\left(-48+\left(n-1\right)\times 4\right)=336\times 2$$

Use the distributive property to multiply n-1 by 4.

$$n\left(-48+4n-4\right)=336\times 2$$

Subtract 4 from -48 to get -52.

$$n\left(-52+4n\right)=336\times 2$$

Use the distributive property to multiply n by -52+4n.

$$-52n+4n^{2}=336\times 2$$

Multiply 336 and 2 to get 672.

$$-52n+4n^{2}=672$$

Subtract 672 from both sides.

$$-52n+4n^{2}-672=0$$

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.

$$4n^{2}-52n-672=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -52 for b, and -672 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 4\left(-672\right)}}{2\times 4}$$

Square -52.

$$n=\frac{-\left(-52\right)±\sqrt{2704-4\times 4\left(-672\right)}}{2\times 4}$$

Multiply -4 times 4.

$$n=\frac{-\left(-52\right)±\sqrt{2704-16\left(-672\right)}}{2\times 4}$$

Multiply -16 times -672.

$$n=\frac{-\left(-52\right)±\sqrt{2704+10752}}{2\times 4}$$

Add 2704 to 10752.

$$n=\frac{-\left(-52\right)±\sqrt{13456}}{2\times 4}$$

Take the square root of 13456.

$$n=\frac{-\left(-52\right)±116}{2\times 4}$$

The opposite of -52 is 52.

$$n=\frac{52±116}{2\times 4}$$

Multiply 2 times 4.

$$n=\frac{52±116}{8}$$

Now solve the equation n=\frac{52±116}{8} when ± is plus. Add 52 to 116.

$$n=\frac{168}{8}$$

Divide 168 by 8.

$$n=21$$

Now solve the equation n=\frac{52±116}{8} when ± is minus. Subtract 116 from 52.

$$n=-\frac{64}{8}$$

Divide -64 by 8.

$$n=-8$$

The equation is now solved.

$$n=21 n=-8$$

Steps for Completing the Square

Multiply both sides by 2.

$$n\left(-48+\left(n-1\right)\times 4\right)=336\times 2$$

Use the distributive property to multiply n-1 by 4.

$$n\left(-48+4n-4\right)=336\times 2$$

Subtract 4 from -48 to get -52.

$$n\left(-52+4n\right)=336\times 2$$

Use the distributive property to multiply n by -52+4n.

$$-52n+4n^{2}=336\times 2$$

Multiply 336 and 2 to get 672.

$$-52n+4n^{2}=672$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$4n^{2}-52n=672$$

Divide both sides by 4.

$$\frac{4n^{2}-52n}{4}=\frac{672}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$n^{2}+\left(-\frac{52}{4}\right)n=\frac{672}{4}$$

Divide -52 by 4.

$$n^{2}-13n=\frac{672}{4}$$

Divide 672 by 4.

$$n^{2}-13n=168$$

Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-13n+\left(-\frac{13}{2}\right)^{2}=168+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-13n+\frac{169}{4}=168+\frac{169}{4}$$

Add 168 to \frac{169}{4}.

$$n^{2}-13n+\frac{169}{4}=\frac{841}{4}$$

Factor n^{2}-13n+\frac{169}{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}.

$$\left(n-\frac{13}{2}\right)^{2}=\frac{841}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{13}{2}\right)^{2}}=\sqrt{\frac{841}{4}}$$

Simplify.

$$n-\frac{13}{2}=\frac{29}{2} n-\frac{13}{2}=-\frac{29}{2}$$

Add \frac{13}{2} to both sides of the equation.

$$n=21 n=-8$$