Answer:
[tex]y=4x^2-8x-4[/tex]
Step-by-step explanation:
Quadratic Function
The quadratic function can be expressed in the following form:
[tex]y=a(x-x_1)(x-x_2)[/tex]
Where a is a real number different from 0, and x1, x2 are the roots or zeroes of the function.
From the conditions stated in the problem, we know
x_1=1+\sqrt{2}, \ x_1=1-\sqrt{2}
Substitute in the general formula above:
[tex]y=a[x-(1+\sqrt{2})][x-(1-\sqrt{2})][/tex]
Operate the indicated product
[tex]y=a(x^2-2x-1)[/tex]
To find the value of a, we use the y-intercept which is the value of y when x=0, thus
[tex]y=a(0^2-2(0)-1)=-4[/tex]
It follows that
[tex]a=4[/tex]
Thus, the required quadratic function is
[tex]y=4(x^2-2x-1)[/tex]
Or, equivalently
[tex]\boxed{y=4x^2-8x-4}[/tex]
