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