Solve 175=n(4+3n+1)/2

$$175 = \frac { n ( 4 + 3 n + 1 ) } { 2 }$$

Solve for n

$$n = -\frac{35}{3} = -11\frac{2}{3} \approx -11.666666667$$

$$n=10$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$175\times 2=n\left(4+3n+1\right)$$

Multiply 175 and 2 to get 350.

$$350=n\left(4+3n+1\right)$$

Add 4 and 1 to get 5.

$$350=n\left(5+3n\right)$$

Use the distributive property to multiply n by 5+3n.

$$350=5n+3n^{2}$$

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

$$5n+3n^{2}=350$$

Subtract 350 from both sides.

$$5n+3n^{2}-350=0$$

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.

$$3n^{2}+5n-350=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 5 for b, and -350 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-5±\sqrt{5^{2}-4\times 3\left(-350\right)}}{2\times 3}$$

Square 5.

$$n=\frac{-5±\sqrt{25-4\times 3\left(-350\right)}}{2\times 3}$$

Multiply -4 times 3.

$$n=\frac{-5±\sqrt{25-12\left(-350\right)}}{2\times 3}$$

Multiply -12 times -350.

$$n=\frac{-5±\sqrt{25+4200}}{2\times 3}$$

Add 25 to 4200.

$$n=\frac{-5±\sqrt{4225}}{2\times 3}$$

Take the square root of 4225.

$$n=\frac{-5±65}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{-5±65}{6}$$

Now solve the equation n=\frac{-5±65}{6} when ± is plus. Add -5 to 65.

$$n=\frac{60}{6}$$

Divide 60 by 6.

$$n=10$$

Now solve the equation n=\frac{-5±65}{6} when ± is minus. Subtract 65 from -5.

$$n=-\frac{70}{6}$$

Reduce the fraction \frac{-70}{6} to lowest terms by extracting and canceling out 2.

$$n=-\frac{35}{3}$$

The equation is now solved.

$$n=10 n=-\frac{35}{3}$$

Steps for Completing the Square

Multiply both sides by 2.

$$175\times 2=n\left(4+3n+1\right)$$

Multiply 175 and 2 to get 350.

$$350=n\left(4+3n+1\right)$$

Add 4 and 1 to get 5.

$$350=n\left(5+3n\right)$$

Use the distributive property to multiply n by 5+3n.

$$350=5n+3n^{2}$$

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

$$5n+3n^{2}=350$$

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.

$$3n^{2}+5n=350$$

Divide both sides by 3.

$$\frac{3n^{2}+5n}{3}=\frac{350}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$n^{2}+\frac{5}{3}n=\frac{350}{3}$$

Divide \frac{5}{3}, the coefficient of the x term, by 2 to get \frac{5}{6}. Then add the square of \frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+\frac{5}{3}n+\left(\frac{5}{6}\right)^{2}=\frac{350}{3}+\left(\frac{5}{6}\right)^{2}$$

Square \frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+\frac{5}{3}n+\frac{25}{36}=\frac{350}{3}+\frac{25}{36}$$

Add \frac{350}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}+\frac{5}{3}n+\frac{25}{36}=\frac{4225}{36}$$

Factor n^{2}+\frac{5}{3}n+\frac{25}{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}.

$$\left(n+\frac{5}{6}\right)^{2}=\frac{4225}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{5}{6}\right)^{2}}=\sqrt{\frac{4225}{36}}$$

Simplify.

$$n+\frac{5}{6}=\frac{65}{6} n+\frac{5}{6}=-\frac{65}{6}$$

Subtract \frac{5}{6} from both sides of the equation.

$$n=10 n=-\frac{35}{3}$$