Solve n^2+30*151n-12060*69=0

$$n ^ { 2 } + 30 \cdot 151 n - 12060 \cdot 69 = 0$$

Solve for n

$$n=3\sqrt{662485}-2265\approx 176.795445978$$

$$n=-3\sqrt{662485}-2265\approx -4706.795445978$$

Steps Using the Quadratic Formula

Multiply 30 and 151 to get 4530.

$$n^{2}+4530n-12060\times 69=0$$

Multiply 12060 and 69 to get 832140.

$$n^{2}+4530n-832140=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4530 for b, and -832140 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-4530±\sqrt{4530^{2}-4\left(-832140\right)}}{2}$$

Square 4530.

$$n=\frac{-4530±\sqrt{20520900-4\left(-832140\right)}}{2}$$

Multiply -4 times -832140.

$$n=\frac{-4530±\sqrt{20520900+3328560}}{2}$$

Add 20520900 to 3328560.

$$n=\frac{-4530±\sqrt{23849460}}{2}$$

Take the square root of 23849460.

$$n=\frac{-4530±6\sqrt{662485}}{2}$$

Now solve the equation n=\frac{-4530±6\sqrt{662485}}{2} when ± is plus. Add -4530 to 6\sqrt{662485}.

$$n=\frac{6\sqrt{662485}-4530}{2}$$

Divide -4530+6\sqrt{662485} by 2.

$$n=3\sqrt{662485}-2265$$

Now solve the equation n=\frac{-4530±6\sqrt{662485}}{2} when ± is minus. Subtract 6\sqrt{662485} from -4530.

$$n=\frac{-6\sqrt{662485}-4530}{2}$$

Divide -4530-6\sqrt{662485} by 2.

$$n=-3\sqrt{662485}-2265$$

The equation is now solved.

$$n=3\sqrt{662485}-2265 n=-3\sqrt{662485}-2265$$

Steps for Completing the Square

Multiply 30 and 151 to get 4530.

$$n^{2}+4530n-12060\times 69=0$$

Multiply 12060 and 69 to get 832140.

$$n^{2}+4530n-832140=0$$

Add 832140 to both sides. Anything plus zero gives itself.

$$n^{2}+4530n=832140$$

Divide 4530, the coefficient of the x term, by 2 to get 2265. Then add the square of 2265 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+4530n+2265^{2}=832140+2265^{2}$$

Square 2265.

$$n^{2}+4530n+5130225=832140+5130225$$

Add 832140 to 5130225.

$$n^{2}+4530n+5130225=5962365$$

Factor n^{2}+4530n+5130225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+2265\right)^{2}=5962365$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+2265\right)^{2}}=\sqrt{5962365}$$

Simplify.

$$n+2265=3\sqrt{662485} n+2265=-3\sqrt{662485}$$

Subtract 2265 from both sides of the equation.

$$n=3\sqrt{662485}-2265 n=-3\sqrt{662485}-2265$$