Answer :
the x-intercepts are 3 and 4
The vertex is the lowest point on the curve
we know that
The equation of a vertical parabola in vertex form is equal to
y = a [tex](x-h)^{2}[/tex] + k
where
(h,k) is the vertex of the parabola
if a>0 -----> then the parabola opens upward (a vertex is a minimum)
if a<0 -----> then the parabola opens downward (a vertex is a maximum)
In this problem we have
y = [tex]x^{2}[/tex] - 7x +12
a = 1
so
the parabola opens upward (a vertex is a minimum)
Find the x-intercepts of the quadratic equation
Equate the equation to zero
y = [tex]x^{2}[/tex] - 7x +12
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]x^{2}[/tex] - 7x = -12
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]x^{2}[/tex] - 7x + 12.25 = -12 + 12.25
[tex]x^{2}[/tex] - 7x + 12.25 = 0.25
Rewrite as perfect squares
[tex](x-3.5)^{2}[/tex] = 0.25
square root on both sides
(x - 3.5) = (+/-) 0.5
x =3.5(+/-) 0.5
x = 3.5+0.5=4
x = 3.5-0.5=3
therefore
the x-intercepts are 3 and 4
A parabola is an approximately U-shaped, mirror-symmetrical plane curve in mathematics. It corresponds to a variety of seemingly unrelated mathematical descriptions, all of which may be shown to define the same curves.
One definition of a parabola includes a line and some extent (the focus) (the directrix). The directrix isn't the main focus. The locus of points therein plane that is equally spaced apart from the directrix and the focus is known as the parabola. A right circular conical surface and a plane parallel to a different plane that is tangential to the conical surface intersect to form a parabola, which is additionally known as a conic section.
To learn more about parabola from the given link:
brainly.com/question/21685473
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