By definition of the zeros of a quadratic function, the time the ball reached the goal is 0.1712.
Zeros of a function
The points where a polynomial function crosses the axis of the independent term (x) represent the so-called zeros of the function.
That is, the zeros represent the roots of the polynomial equation that is obtained by making f(x)=0.
In summary, the roots or zeros of the quadratic function are those values of x for which the expression is equal to 0. Graphically, the roots correspond to the abscissa of the points where the parabola intersects the x-axis.
In a quadratic function that has the form:
f(x)= ax² + bx + c
the zeros or roots are calculated by:
[tex]x1,x2=\frac{-b+-\sqrt{b^{2}-4ac } }{2a}[/tex]
Time the ball reached the goal
The equation for the height h of the ball at time t is h=-4.9t²-5t+2.
A soccer player kicks the ball to a height of 1 meter inside the goal. This is, h=1. Replacing:
1=-4.9t²-5t+2
Solving:
0=-4.9t²-5t+2 -1
0=-4.9t²-5t+1
So, being:
the zeros or roots are calculated as:
[tex]t1=\frac{-(-5)+\sqrt{(-5)^{2}-4x(-4.9)x1 } }{2x(-4.9)}[/tex]
[tex]t1=\frac{5+\sqrt{25 + 19.6 } }{-9.8}[/tex]
[tex]t1=\frac{5+\sqrt{44.6 } }{-9.8}[/tex]
[tex]t1=\frac{5+6.678 }{-9.8}[/tex]
[tex]t1=\frac{11.678 }{-9.8}[/tex]
t1= -1.192
and
[tex]t2=\frac{-(-5)-\sqrt{(-5)^{2}-4x(-4.9)x1 } }{2x(-4.9)}[/tex]
[tex]t2=\frac{5-\sqrt{25 + 19.6 } }{-9.8}[/tex]
[tex]t2=\frac{5-\sqrt{44.6 } }{-9.8}[/tex]
[tex]t2=\frac{5-6.678 }{-9.8}[/tex]
[tex]t2=\frac{-1.678}{-9.8}[/tex]
t2= 0.1712
Since time cannot be negative, the time the ball reached the goal is 0.1712.
Learn more about the zeros of a quadratic function:
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