Answer:
[tex](x-2)^2=36[/tex]
Step-by-step explanation:
In order to figure out what number should go in the blank within the parentheses, we use the formula [tex]\frac{b}{2}[/tex] , where b is the coefficient of the x term. In this case, b = -4. So, the number in the parentheses blank is: [tex]\frac{-4}{2} =-2[/tex].
Now, on the left, our parenthetical expression is: [tex](x-2)^2[/tex]. However, we're not done. The top and bottom equations still need to equal each other. Let's see if they are equal by expanding the [tex](x-2)^2[/tex]: [tex](x-2)^2=x^2-4x+4[/tex]. Clearly, they're not the same because the top equation has a -32, whereas the bottom one has +4. So, in order to make these the same, we need to add or subtract something to the right side of the bottom equation.
What will make the left side have -32? If the right side has + 36: [tex]x^2-4x+4=36[/tex]. We can check this by subtracting 36 from both sides: [tex]x^2-4x+4-36=36-36[/tex] ⇒ [tex]x^2-4x-32=0[/tex] ... and they match.
So, our final answer is: [tex](x-2)^2=36[/tex]
Hope this helps!
