Solve n[-34+(n-1)*3]/2=230

$$\frac { n [ - 34 + ( n - 1 ) \times 3 ] } { 2 } = 230$$

Solve for n

$$n = -\frac{23}{3} = -7\frac{2}{3} \approx -7.666666667$$

$$n=20$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$n\left(-34+\left(n-1\right)\times 3\right)=230\times 2$$

Use the distributive property to multiply n-1 by 3.

$$n\left(-34+3n-3\right)=230\times 2$$

Subtract 3 from -34 to get -37.

$$n\left(-37+3n\right)=230\times 2$$

Use the distributive property to multiply n by -37+3n.

$$-37n+3n^{2}=230\times 2$$

Multiply 230 and 2 to get 460.

$$-37n+3n^{2}=460$$

Subtract 460 from both sides.

$$-37n+3n^{2}-460=0$$

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.

$$3n^{2}-37n-460=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -37 for b, and -460 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\times 3\left(-460\right)}}{2\times 3}$$

Square -37.

$$n=\frac{-\left(-37\right)±\sqrt{1369-4\times 3\left(-460\right)}}{2\times 3}$$

Multiply -4 times 3.

$$n=\frac{-\left(-37\right)±\sqrt{1369-12\left(-460\right)}}{2\times 3}$$

Multiply -12 times -460.

$$n=\frac{-\left(-37\right)±\sqrt{1369+5520}}{2\times 3}$$

Add 1369 to 5520.

$$n=\frac{-\left(-37\right)±\sqrt{6889}}{2\times 3}$$

Take the square root of 6889.

$$n=\frac{-\left(-37\right)±83}{2\times 3}$$

The opposite of -37 is 37.

$$n=\frac{37±83}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{37±83}{6}$$

Now solve the equation n=\frac{37±83}{6} when ± is plus. Add 37 to 83.

$$n=\frac{120}{6}$$

Divide 120 by 6.

$$n=20$$

Now solve the equation n=\frac{37±83}{6} when ± is minus. Subtract 83 from 37.

$$n=-\frac{46}{6}$$

Reduce the fraction \frac{-46}{6} to lowest terms by extracting and canceling out 2.

$$n=-\frac{23}{3}$$

The equation is now solved.

$$n=20 n=-\frac{23}{3}$$

Steps for Completing the Square

Multiply both sides by 2.

$$n\left(-34+\left(n-1\right)\times 3\right)=230\times 2$$

Use the distributive property to multiply n-1 by 3.

$$n\left(-34+3n-3\right)=230\times 2$$

Subtract 3 from -34 to get -37.

$$n\left(-37+3n\right)=230\times 2$$

Use the distributive property to multiply n by -37+3n.

$$-37n+3n^{2}=230\times 2$$

Multiply 230 and 2 to get 460.

$$-37n+3n^{2}=460$$

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.

$$3n^{2}-37n=460$$

Divide both sides by 3.

$$\frac{3n^{2}-37n}{3}=\frac{460}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$n^{2}-\frac{37}{3}n=\frac{460}{3}$$

Divide -\frac{37}{3}, the coefficient of the x term, by 2 to get -\frac{37}{6}. Then add the square of -\frac{37}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{37}{3}n+\left(-\frac{37}{6}\right)^{2}=\frac{460}{3}+\left(-\frac{37}{6}\right)^{2}$$

Square -\frac{37}{6} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{37}{3}n+\frac{1369}{36}=\frac{460}{3}+\frac{1369}{36}$$

Add \frac{460}{3} to \frac{1369}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{37}{3}n+\frac{1369}{36}=\frac{6889}{36}$$

Factor n^{2}-\frac{37}{3}n+\frac{1369}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{37}{6}\right)^{2}=\frac{6889}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{37}{6}\right)^{2}}=\sqrt{\frac{6889}{36}}$$

Simplify.

$$n-\frac{37}{6}=\frac{83}{6} n-\frac{37}{6}=-\frac{83}{6}$$

Add \frac{37}{6} to both sides of the equation.

$$n=20 n=-\frac{23}{3}$$