Question 1202224
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Answer:  <font color=red size=4>x^2 - 10x + 16 = 0</font>
Other equations are possible


Explanation:


The term "zero" of a function is the same as a root.
For real numbers, the root is the x intercept.


The given roots are 8 and 2
x = 8 leads to x-8 = 0, so (x-8) is one factor
x = 2 leads to (x-2) being the other factor


We then need to expand out (x-8)(x-2)


We could use the FOIL rule to expand it out, but I'll use the <a href="https://www.algebra.com/algebra/homework/playground/lessons/box-method.lesson">box method</a>


First place the terms along the left and top edge like so
<table border = "1" cellpadding = "5"><tr><td></td><td>x</td><td>-8</td></tr><tr><td>x</td><td></td><td></td></tr><tr><td>-2</td><td></td><td></td></tr></table>


Then fill each box with the product of the headers
Example: top left corner is x^2 because x*x = x^2
Another example: bottom right corner is 16 because -2*(-8) = 16


This is what the table would look like when everything is filled in
<table border = "1" cellpadding = "5"><tr><td></td><td>x</td><td>-8</td></tr><tr><td>x</td><td>x^2</td><td>-8x</td></tr><tr><td>-2</td><td>-2x</td><td>16</td></tr></table>


Then we add up the terms. 
Combine like terms if possible
x^2 + (-8x) + (-2x) + 16
<font color="red">x^2 - 10x + 16</font>


Therefore, (x-8)(x-2) = <font color="red">x^2 - 10x + 16</font>



The equation
<font color="red">x^2 - 10x + 16 = 0</font>
leads to the roots x = 8 and x = 2
Other equations lead to these roots because we can scale up or down the equation. For instance, triple both sides to go from x^2-10x+16 = 0 to 3x^2-30x+48 = 0. Both of those equations have the same roots.


Check:
Let's try x = 8
x^2 - 10x + 16 = 0
8^2 - 10*8 + 16 = 0
64 - 10*8 + 16 = 0
64 - 80 + 16 = 0
-16 + 16 = 0
0 = 0
This confirms x = 8 as a root
I'll let you try x = 2. 
The goal is to get the left hand side to be zero.


Another way to verify is to use a graphing tool like Desmos as shown here
<a href="https://www.desmos.com/calculator/gcibt9azif">https://www.desmos.com/calculator/gcibt9azif</a>
The parabola has x intercepts 2 and 8 where the graph crosses the x axis.
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