# The attendances y for two movies can be modeled by the following equations, where x is the number of days since the movies opened. Movie A: y=-x^{2}+35x+100 Movie B: y=-5x+275 Where x is number of days since the movies opened. When is the attendance for each movie the same?

The attendances y for two movies can be modeled by the following equations, where x is the number of days since the movies opened.
Movie A: $$\displaystyle{y}=-{x}^{{{2}}}+{35}{x}+{100}$$
Movie B: $$\displaystyle{y}=-{5}{x}+{275}$$
Where x is number of days since the movies opened.
When is the attendance for each movie the same?

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Arnold Odonnell
Formula used:
$$\displaystyle{x}={\frac{{-{b}\pm\sqrt{{{b}^{{{2}}}-{4}{a}{c}}}}}{{{2}{a}}}}$$
Calculation:
Substituting value of equation 1 in equation 2,
$$\displaystyle-{x}^{{{2}}}+{35}{x}+{100}=-{5}{x}+{275}$$
Shifting the value,
$$\displaystyle-{x}^{{{2}}}+{35}{x}+{100}=-{5}{x}+{275}={0}$$
On solving,
$$\displaystyle-{x}^{{{2}}}+{40}{x}-{175}={0}$$
Hence,
$$\displaystyle{a}=-{1},$$
$$\displaystyle{b}={40},$$
$$\displaystyle{c}=-{175},$$
Substituting the formula
$$\displaystyle{x}={\frac{{-{40}\pm\sqrt{{{40}^{{{2}}}-{4}{\left(-{1}\right)}{\left(-{175}\right)}}}}}{{{2}{\left(-{1}\right)}}}}$$
So,
$$\displaystyle{x}={\frac{{-{40}\pm\sqrt{{{1600}-{700}}}}}{{-{2}}}}$$
$$\displaystyle{x}={\frac{{-{40}\pm\sqrt{{{900}}}}}{{-{2}}}}$$
$$\displaystyle{x}={\frac{{-{40}\pm{30}}}{{-{2}}}}$$
$$\displaystyle{x}={\frac{{-{40}+{30}}}{{-{2}}}}$$
$$\displaystyle{x}={5}$$
Value of y, substituting in equation 2,
$$\displaystyle{y}=-{5}{\left({5}\right)}+{275}$$
$$\displaystyle{y}={250}$$
For -30,
$$\displaystyle{x}={\frac{{-{40}-{30}}}{{-{2}}}}$$
$$\displaystyle{x}={35}$$
Value of y, substituting in equation 2,
$$\displaystyle{y}=-{5}{\left({35}\right)}+{275}$$
$$\displaystyle{y}={100}$$
Hence, $$\displaystyle{x}={5},{35}\ \text{and}\ {y}={250},{100}$$
Hence, the attendance of two movies is same on day 5 and day 35