# Find the arithmetic mean of the roots of the equation √ (2x + 8) -√ (x + 2) = 2

The equation is given

sqrt {x + 2} + sqrt {2 x + 8} = 2

Let’s raise both parts of the ur-niya in (o) 2nd degree

(- sqrt {x + 2} + sqrt {2 x + 8}) ^ {2} = 4

or

1 ^ {2} (2 x + 8) + – 2 sqrt {(x + 2) (2 x + 8)} + (-1) ^ {2} (x + 2) = 4

or

3 x – 2sqrt {2 x ^ {2} + 12 x + 16} + 10 = 4

transform:

– 2 sqrt {2 x ^ {2} + 12 x + 16} = – 3 x – 6

Let’s raise both parts of the ur-niya in (o) 2nd degree

8 x ^ {2} + 48 x + 64 = (- 3 x – 6) ^ {2}

8 x ^ {2} + 48 x + 64 = 9 x ^ {2} + 36 x + 36

Move the right side of the equation to the left side of the equation with a minus sign

– x ^ {2} + 12 x + 28 = 0

This is an equation of the form

a * x ^ 2 + b * x + c = 0

The quadratic equation can be solved using the discriminant.

D = b ^ 2 – 4 * a * c

Because

a = -1

b = 12

c = 28

, then

D = b ^ 2 – 4 * a * c = (12) ^ 2 – 4 * (-1) * (28) = 256

Because D> 0, then the equation has two roots.

x1 = (-b + sqrt (D)) / (2 * a)

x2 = (-b – sqrt (D)) / (2 * a)

or

x_ {1} = -2

x_ {2} = 14

Because

sqrt {2 x ^ {2} + 12 x + 16} = {3 x} {2} + 3

and

sqrt {2 x ^ {2} + 12 x + 16}> 0

then

{3 x} / {2} + 3> 0

or

x_ {1} = -2

x_ {2} = 14

check:

x_ {1} = -2

– sqrt {x_ {1} + 2} + sqrt {2 x_ {1} + 8} – 2 = 0

=

-2 + – sqrt {-2 + 2} + sqrt {-2/2 + 8} = 0

=

0 = 0 – identity

x_ {2} = 14

– sqrt {x_ {2} + 2} + sqrt {2 x_ {2} + 8} – 2 = 0

=

-2 + – sqrt {2 + 14} + sqrt {8 + 2/14} = 0

=

0 = 0

– identity

Then, the final answer is:

x_ {1} = -2

x_ {2} = 14

Average:

((-2) + 14) / 2 = 6