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.

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

.