Respuesta :
Answer:
the points (-5,0) and (3,0) are the x-intercepts of the parabola.
Step-by-step explanation:
The equation of a parabola with vertex in (h,k) is:
[tex](x-h)^2=4p(y-k)[/tex]
Where p is a constant. Replacing the (h,k) by (-1,80), we get:
[tex](x+1)^2=4p(y-80)[/tex]
Additionally, we know that when x is 0, y is equal to 75. So, we can replace these values and solve for p as:
[tex](x+1)^2=4p(y-80)\\(0+1)^2=4p(75-80)\\1=4p(-5)\\p=\frac{-1}{20}[/tex]
So, replacing the value of p on the initial equation, we get that the equation of the parabola is equal to:
[tex](x+1)^2=4(\frac{-1}{20})(y-80)\\(x+1)^2=\frac{-1}{5} (y-80)[/tex]
Solving for y, we get:
[tex]x^2+2x+1=\frac{-1}{5}y+16 \\x^2+2x+1-16=\frac{-1}{5}y\\-5x^2-10x+75=y[/tex]
Then, the x-intercepts of the parabola are the values of x where y is equal to zero, so we need to solving the following equation:
[tex]-5x^2-10x+75=0[/tex]
So, the values of x are calculated as:
[tex]x_1=\frac{10+\sqrt{10^2-4(-5)(75)} }{2(-5)}=-5\\x_1=\frac{10-\sqrt{10^2-4(-5)(75)} }{2(-5)}=3\\[/tex]
Finally, the points (-5,0) and (3,0) are the x-intercepts of the parabola.
