Step-by-step explanation:
Proving by Vertex Form:Yes, if we put this equation in vertex form
[tex]a(x - h) {}^{2} + k[/tex]
where h,k is vertex
we can find if it reaches 5.
First we do -b/2a.
[tex] \frac{ - 1}{10} [/tex]
Is a solution, if we plug that in we get
Which gives
[tex]3.95[/tex]
So the vertex form is
[tex]5(x + 0.1) {}^{2} + 3.95[/tex]
Since the graph is facing upwards,because a is positive, the function will pass all values greater than 3.95, and 5 is one of them
Second Way: Watch if you want to see how long is reaches 5 meters.
Set the quadratic equation equal to 5.
[tex]5 {t}^{2} + t + 4 = 5[/tex]
[tex] {5t}^{2} + t - 1 = 0[/tex]
[tex]5 {t}^{2} + t - 1 = 0[/tex]
Do quadratic formula, which is
[tex] - b± \frac{ \sqrt{b {}^{2} - 4ac} }{2a} [/tex]
So a is 5 b is 1 and c is -1 so we get
[tex] - 1± \frac{ \sqrt{1 {}^{2} - 4( 5)( - 1)} }{2(5)} [/tex]
[tex] - 1± \frac{ \sqrt{1 + 20} }{10} [/tex]
[tex] - 1± \frac{ \sqrt{21} }{10} [/tex]
[tex] \frac{ - 1}{10} + \frac{ \sqrt{21} }{10} [/tex]
and
[tex] \frac{ - 1}{10} - \frac{ \sqrt{21} }{10} [/tex]
are the solutions but since time is scalar, it can't be negative so qe take the positive solution
[tex] \frac{ - 1}{10} + \frac{ \sqrt{21} }{10} [/tex]
which is approximate 0.358.
So if you wanted to know, it will take 0.36 seconds to reach a height of 5 meters.
