Solve 75n=68n-n^2+cos544-8n

$$75 n = 68 n - n ^ { 2 } + \cos 544 - 8 n$$

Solve for n

$$n=\frac{\sqrt{552524359497401758}}{100000000}-7.5\approx -0.066801768$$

$$n=-\frac{\sqrt{552524359497401758}}{100000000}-7.5\approx -14.933198232$$

Steps Using the Quadratic Formula

Evaluate trigonometric functions in the problem

$$75 n = 68 n - n ^ {2} + -0.9975640502598242 - 8 n$$

Combine 68n and -8n to get 60n.

$$75n=60n-n^{2}-0.9975640502598242$$

Subtract 60n from both sides.

$$75n-60n=-n^{2}-0.9975640502598242$$

Combine 75n and -60n to get 15n.

$$15n=-n^{2}-0.9975640502598242$$

Add n^{2} to both sides.

$$15n+n^{2}=-0.9975640502598242$$

Add 0.9975640502598242 to both sides.

$$15n+n^{2}+0.9975640502598242=0$$

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.

$$n^{2}+15n+0.9975640502598242=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 15 for b, and 0.9975640502598242 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-15±\sqrt{15^{2}-4\times 0.9975640502598242}}{2}$$

Square 15.

$$n=\frac{-15±\sqrt{225-4\times 0.9975640502598242}}{2}$$

Multiply -4 times 0.9975640502598242.

$$n=\frac{-15±\sqrt{225-3.9902562010392968}}{2}$$

Add 225 to -3.9902562010392968.

$$n=\frac{-15±\sqrt{221.0097437989607032}}{2}$$

Take the square root of 221.0097437989607032.

$$n=\frac{-15±\frac{\sqrt{552524359497401758}}{50000000}}{2}$$

Now solve the equation n=\frac{-15±\frac{\sqrt{552524359497401758}}{50000000}}{2} when ± is plus. Add -15 to \frac{\sqrt{552524359497401758}}{50000000}.

$$n=\frac{\frac{\sqrt{552524359497401758}}{50000000}-15}{2}$$

Divide -15+\frac{\sqrt{552524359497401758}}{50000000} by 2.

$$n=\frac{\sqrt{552524359497401758}}{100000000}-\frac{15}{2}$$

Now solve the equation n=\frac{-15±\frac{\sqrt{552524359497401758}}{50000000}}{2} when ± is minus. Subtract \frac{\sqrt{552524359497401758}}{50000000} from -15.

$$n=\frac{-\frac{\sqrt{552524359497401758}}{50000000}-15}{2}$$

Divide -15-\frac{\sqrt{552524359497401758}}{50000000} by 2.

$$n=-\frac{\sqrt{552524359497401758}}{100000000}-\frac{15}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{552524359497401758}}{100000000}-\frac{15}{2} n=-\frac{\sqrt{552524359497401758}}{100000000}-\frac{15}{2}$$

Steps for Completing the Square

Evaluate trigonometric functions in the problem

$$75 n = 68 n - n ^ {2} + -0.9975640502598242 - 8 n$$

Combine 68n and -8n to get 60n.

$$75n=60n-n^{2}-0.9975640502598242$$

Subtract 60n from both sides.

$$75n-60n=-n^{2}-0.9975640502598242$$

Combine 75n and -60n to get 15n.

$$15n=-n^{2}-0.9975640502598242$$

Add n^{2} to both sides.

$$15n+n^{2}=-0.9975640502598242$$

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.

$$n^{2}+15n=-0.9975640502598242$$

Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+15n+\left(\frac{15}{2}\right)^{2}=-0.9975640502598242+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+15n+\frac{225}{4}=-0.9975640502598242+\frac{225}{4}$$

Add -0.9975640502598242 to \frac{225}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}+15n+\frac{225}{4}=\frac{276262179748700879}{5000000000000000}$$

Factor n^{2}+15n+\frac{225}{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}.

$$\left(n+\frac{15}{2}\right)^{2}=\frac{276262179748700879}{5000000000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{15}{2}\right)^{2}}=\sqrt{\frac{276262179748700879}{5000000000000000}}$$

Simplify.

$$n+\frac{15}{2}=\frac{\sqrt{552524359497401758}}{100000000} n+\frac{15}{2}=-\frac{\sqrt{552524359497401758}}{100000000}$$

Subtract \frac{15}{2} from both sides of the equation.

$$n=\frac{\sqrt{552524359497401758}}{100000000}-\frac{15}{2} n=-\frac{\sqrt{552524359497401758}}{100000000}-\frac{15}{2}$$