Respuesta :
Answer:
-1.46 and 0.88
Step-by-step explanation:
Formula of the Vertex of Parabola y = ax² + bx + c :
[tex]\boxed{\left(-\frac{b}{2a} ,\ -\frac{b^2-4ac}{4a}\right) }[/tex]
Since, the y-intercept = -9, then c = -9
*Proof: by substituting (0, -9) into y = ax² + bx + c
-9 = a(0)² + b(0) + c
[tex]\boxed{\bf c=-9}[/tex]
The Vertex = [tex]\displaystyle\left(-\frac{2}{7} ,\ -9\frac{4}{7} \right)[/tex], which means:
- [tex]\displaystyle -\frac{b}{2a} =-\frac{2}{7}[/tex]
- [tex]\displaystyle -\frac{b^2-4ac}{4a} =-9\frac{4}{7}[/tex]
We start with the 1st equation:
[tex]\displaystyle -\frac{b}{2a} =-\frac{2}{7}[/tex]
[tex]7b=2(2a)[/tex]
[tex]\displaystyle b=\frac{4a}{7} \ ...\ [1][/tex]
Now, substitute [1] & c-value to the 2nd equation:
[tex]\displaystyle -\frac{b^2-4ac}{4a} =-9\frac{4}{7}[/tex]
[tex]\displaystyle \frac{(\frac{4a}{7} )^2-4a(-9)}{4a} =\frac{67}{7}[/tex]
[tex]\displaystyle 7\left(\frac{16a^2}{49} +36a\right) =67(4a)[/tex]
[tex]\displaystyle \frac{16a^2}{7} +252a =268a[/tex]
[tex]\displaystyle \frac{16a^2}{7}=268a-252a[/tex]
[tex]16a^2=7(16a)[/tex]
[tex]\boxed{\bf a= 7}[/tex]
Substitute a-value with 7 to the 1st equation to find the b-value:
[tex]\displaystyle b=\frac{4a}{7}[/tex]
[tex]\displaystyle b=\frac{4(7)}{7}[/tex]
[tex]\boxed{\bf b=4}[/tex]
With the value of a, b and c, we can find the roots / zeros of the parabola using the Quadratic Roots Formula:
[tex]\boxed{x=\frac{-b\pm\sqrt{b^2-4ac} }{2a} }[/tex]
[tex]\displaystyle x=\frac{-4\pm\sqrt{4^2-4(7)(-9)} }{2(7)}[/tex]
[tex]\boxed{\bf x=-1.46\ and\ 0.88}[/tex]
