Answer:
[tex]y = 3(x + 4.5)(x + 2.8)[/tex].
Step-by-step explanation:
Start with the two [tex]x[/tex]-intercepts. The two zeros of the quadratic equation for this parabola are:
(These are the [tex]x[/tex]-coordinates of the two [tex]x[/tex]-intercepts.)
By the factor theorem, [tex]x = k[/tex] (where [tex]k[/tex] is a real number) is a zero of a polynomial if and only if [tex](x - k)[/tex] is a factor of that polynomial.
A quadratic equation is also a polynomial. In this case, the two zeros would correspond to the two factors
A parabola could only have up to two factors. As a result, the power of these two factor should both be one. Hence, the equation for the parabola would be in the form
[tex]y = a \, (x + 4.5)(x + 2.8)[/tex],
where [tex]a[/tex] is the leading coefficient that still needs to be found. Calculate the value of [tex]a[/tex] using the [tex]y[/tex]-intercept of this parabola. (Any other point on this parabola that is not one of the two [tex]x[/tex]-intercepts would work.)
Since the coordinates of the [tex]y[/tex]-intercept are [tex](0,\, 37.8)[/tex], [tex]x = 0[/tex] and [tex]y = 37.8[/tex]. The equation [tex]y = a \, (x + 4.5)(x + 2.8)[/tex] becomes:
[tex]37.8 = a \, (0 + 4.5)(0 + 2.8)[/tex].
Solve for [tex]a[/tex]:
[tex]\displaystyle a = \frac{37.8}{4.5\times 2.8} = 3[/tex].
Hence the equation for this parabola:
[tex]y = 3(x + 4.5)(x + 2.8)[/tex].
