The x-intercepts of the given quadratic equation are:
(13, 0) and (-1, 0).
For a quadratic equation with leading coefficient a and vertex (h, k), the quadratic equation can be written as:
[tex]y = a*(x - h)^2 + k[/tex]
Here we know that the vertex is (6, -245), then we can write:
[tex]y = a*(x - 6)^2 - 245[/tex]
And we know that the y-intercept is y = -65, then if we evaluate in 0 we get:
[tex]-65= a*(0 - 6)^2 - 245\\\\-65 = a*36 - 245\\\\-65 + 245 = a*36\\\\180 = a*36\\\\180/36 = a = 5[/tex]
So the quadratic equation is:
[tex]y = 5*(x - 6)^2 - 245[/tex]
Expanding it, we get:
[tex]y = 5x^2 - 60x - 65[/tex]
To get the x-intercepts, we can use Bhaskara's formula:
[tex]x = \frac{60 \pm \sqrt{(-60)^2 - 4*5*(-65)} }{2*5} \\\\x = \frac{60 \pm 70}{10}[/tex]
So we get:
x = (60 + 70)/10 = 13
x= (60 - 70)/10 = -1
The x-intercepts are (13, 0) and (-1, 0).
If you want to learn more about quadratic equations:
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