Solve n(n+1)/2=1225

$$\frac { n ( n + 1 ) } { 2 } = 1225$$

Solve for n

$$n=-50$$

$$n=49$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$n\left(n+1\right)=1225\times 2$$

Use the distributive property to multiply n by n+1.

$$n^{2}+n=1225\times 2$$

Multiply 1225 and 2 to get 2450.

$$n^{2}+n=2450$$

Subtract 2450 from both sides.

$$n^{2}+n-2450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -2450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-2450\right)}}{2}$$

Square 1.

$$n=\frac{-1±\sqrt{1-4\left(-2450\right)}}{2}$$

Multiply -4 times -2450.

$$n=\frac{-1±\sqrt{1+9800}}{2}$$

Add 1 to 9800.

$$n=\frac{-1±\sqrt{9801}}{2}$$

Take the square root of 9801.

$$n=\frac{-1±99}{2}$$

Now solve the equation n=\frac{-1±99}{2} when ± is plus. Add -1 to 99.

$$n=\frac{98}{2}$$

Divide 98 by 2.

$$n=49$$

Now solve the equation n=\frac{-1±99}{2} when ± is minus. Subtract 99 from -1.

$$n=-\frac{100}{2}$$

Divide -100 by 2.

$$n=-50$$

The equation is now solved.

$$n=49 n=-50$$

Steps for Completing the Square

Multiply both sides by 2.

$$n\left(n+1\right)=1225\times 2$$

Use the distributive property to multiply n by n+1.

$$n^{2}+n=1225\times 2$$

Multiply 1225 and 2 to get 2450.

$$n^{2}+n=2450$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=2450+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+n+\frac{1}{4}=2450+\frac{1}{4}$$

Add 2450 to \frac{1}{4}.

$$n^{2}+n+\frac{1}{4}=\frac{9801}{4}$$

Factor n^{2}+n+\frac{1}{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}.

$$\left(n+\frac{1}{2}\right)^{2}=\frac{9801}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{9801}{4}}$$

Simplify.

$$n+\frac{1}{2}=\frac{99}{2} n+\frac{1}{2}=-\frac{99}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$n=49 n=-50$$