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Verify whether the cones are the same:
(i) [tex]x^2 - 2x \quad x = 2, 0[/tex]



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Express 3x^2 − 27 in the form a(x + b)^2 + c and hence find the maximum value of 3x^2 − 27.
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Response:
Express [tex]3x^2 - 27[/tex] in the form [tex]a(x + b)^2 + c[/tex] and hence find the maximum value of [tex]3x^2 - 27[/tex].



Answer :

GinnyAnswer
Sure! Let's verify the values of the function [tex]\( x^2 - 2x \)[/tex] at [tex]\( x = 2 \)[/tex] and [tex]\( x = 0 \)[/tex].

### Step-by-Step Solution

1. Define the Function:
[tex]\[ f(x) = x^2 - 2x \][/tex]

2. Calculate the Function Value at [tex]\( x = 2 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ f(2) = 2^2 - 2 \cdot 2 \][/tex]
- Perform the calculations:
[tex]\[ = 4 - 4 \][/tex]
- Simplify the result:
[tex]\[ = 0 \][/tex]

3. Calculate the Function Value at [tex]\( x = 0 \)[/tex]:
- Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = 0^2 - 2 \cdot 0 \][/tex]
- Perform the calculations:
[tex]\[ = 0 - 0 \][/tex]
- Simplify the result:
[tex]\[ = 0 \][/tex]

Therefore, the values of the function [tex]\( x^2 - 2x \)[/tex] at [tex]\( x = 2 \)[/tex] and [tex]\( x = 0 \)[/tex] are both:
[tex]\[ f(2) = 0 \][/tex]
[tex]\[ f(0) = 0 \][/tex]

So, the verified values are [tex]\( (0, 0) \)[/tex].

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