# Thread: maximum point of the curve

1. ## maximum point of the curve

The curve $y=-2x^2+8x-3$ has a maximum point at $x=p$,where $p$ is constant.Find the value of $p$

2. Originally Posted by mastermin346
The curve $y=-2x^2+8x-3$ has a maximum point at $x=p$,where $p$ is constant.Find the value of $p$
A parabola opening downward has its maximum value at the vertex- and you can get that by completing the square:
$-2x^2+ 8x- 3= -2(x^2- 4x)- 3= -2(x^2- 4x+ 4- 4)- 3$
$= -2(x^2- 4x+ 4)+ 8- 3= -2(x- 2)^2+ 5$

Since a square is never negative, that will be largest when x-2= 0.
The text passage contains a mathematical question about finding the value of \( p \) where the curve \( y = -2x^2 + 8x - 3 \) has a maximum point. The answer to this question is provided through the process of completing the square, which shows that the maximum point occurs at \( x = 2 \).
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Thread: maximum point of the curve

1. ## maximum point of the curve

The curve $y=-2x^2+8x-3$ has a maximum point at $x=p$,where $p$ is constant.Find the value of $p$

2. Originally Posted by mastermin346
The curve $y=-2x^2+8x-3$ has a maximum point at $x=p$,where $p$ is constant.Find the value of $p$
A parabola opening downward has its maximum value at the vertex- and you can get that by completing the square:
$-2x^2+ 8x- 3= -2(x^2- 4x)- 3= -2(x^2- 4x+ 4- 4)- 3$
$= -2(x^2- 4x+ 4)+ 8- 3= -2(x- 2)^2+ 5$

Since a square is never negative, that will be largest when x-2= 0.
user
The text passage contains a mathematical question about finding the value of \( p \) where the curve \( y = -2x^2 + 8x - 3 \) has a maximum point. The answer to this question is provided through the process of completing the square, which shows that the maximum point occurs at \( x = 2 \).

TYPE: SINGLE_QUESTION_WITH_ANSWER