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Greatest or least value of a quadratic expression by completing the square

This type of question, asked 4 times: CDS 2 2018, CDS 1 2019, CDS 2 2024, CDS 1 2025. All 5 questions below, oldest first.

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  1. Question 1

    CDS 2 2018

    21
    The minimum value of the expression 2x2+5x+52x^2 + 5x + 5 is

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  2. Question 2

    CDS 1 2019

    85
    A real number xx is such that (x−x2)(x - x^2) is maximum. What is xx equal to ?

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  3. Question 3

    CDS 1 2019

    93
    What is the maximum value of the expression 1x2+5x+10\frac{1}{x^2 + 5x + 10} ?

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  4. Question 4

    CDS 2 2024

    1
    A real number xx is such that the sum of the number and four times its square is the least. What is that number ?

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  5. Question 5

    CDS 1 2025

    1
    A real number M is squared to give the value N. What is the minimum value of (M+N)(M + N) ?

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