Answer:
[tex]m-n=2[/tex]
Step-by-step explanation:
Instead of using the standard form, we can use the vertex form of a quadratic equation:
[tex]f(x)=a(x-h)^2+k[/tex]
Where a is the leading coefficient, and (h, k) is our vertex.
Our vertex point is at (2, -4). So, let’s substitute 2 for h and -4 for k:
[tex]f(x)=a(x-2)^2-4[/tex]
Now, we need to determine a.
We know that it passes through the point (4, 12). So, when x is 4, y must be 12. In other words:
[tex]12=a((4)-2)^2-4[/tex]
Solve for a. Subtract within the parentheses:
[tex]12=a(2)^2-4[/tex]
Add 4 to both sides:
[tex]16=a(2)^2[/tex]
Square:
[tex]16=4a[/tex]
Solve:
[tex]a=4[/tex]
Thererfore, the value of a is 4.
So, our function is:
[tex]f(x)=4(x-2)^2-4[/tex]
Now, let’s find our roots. Set the equation to 0 and solve for x:
[tex]0=4(x-2)^2-4[/tex]
[tex]4=4(x-2)^2\\1=(x-2)^2\\x-2=\pm1 \\ x=2\pm1 \\ x=3\text{ or } 1[/tex]
So, our roots are 1 and 3.
The greater root is 3 and the lesser root is 1.
Therefore, m-n, where m>n, is 3-1 or 2.
Our final answer is 2.
