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17 July, 20:03

1. A positive real number is 2 less than another. When 4 times the larger is added to the square of the smaller, the result is 49. Find the numbers.

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Answers (1)
  1. 17 July, 20:57
    0
    x=

    -

    2

    +

    3



    5

    y=

    3



    5

    Step-by-step explanation:

    Positive real number =

    x

    Other number =

    y

    First statement: A positive real number is 2 less than another.

    Therefore

    y

    =

    x

    +

    2

    Second statement: When 4 times the larger is added to the square of the smaller, the result is 49.

    Since the larger number is

    y

    , we know that "4 times the

    larger" means

    4

    y

    .

    The square of the smaller would mean

    x

    2

    .

    If you put them together, it would look like

    4

    y

    +

    x

    2

    =

    49

    .

    Now create a system of equations like the following:

    y

    =

    x

    +

    2

    4

    y

    +

    x

    2

    =

    49

    .

    From the first equation, we can understand the value of

    y

    in terms of

    x

    , which is

    x

    +

    2

    .

    Replace the

    y

    in the second equation with

    x

    +

    2

    .

    4

    (

    x

    +

    2

    )

    +

    x

    2

    =

    49

    Now that we have an equation with only one variable, solve for

    x

    .

    Below is a potential way one can solve this equation:

    1. Expand the bracket.

    4

    x

    +

    8

    +

    x

    2

    =

    49

    2. Form a quadratic equation (

    a

    x

    2

    +

    b

    x

    +

    c

    =

    0

    )

    x

    2

    +

    4

    x

    -

    41

    =

    0

    3, Use

    -

    b

    ±



    b

    2

    -

    4

    a

    c

    2

    (

    a

    is 1,

    b

    is 4,

    c

    is - 41.)

    -

    4

    ±



    4

    2

    -

    4



    (

    -

    41

    )

    2

    -

    4

    ±

    (

    180

    )

    1

    2

    2

    -

    2

    ±

    3



    5

    Since the answer must be a positive real number,

    x

    will be

    -

    2

    +

    3



    5

    If you place

    -

    2

    +

    3



    5

    into

    y

    =

    x

    +

    2

    :

    y

    =

    -

    2

    +

    3



    5

    +

    2

    You will get

    3



    5

    for

    y

    .
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