How to find the real solutions of the equation square root(2x+8)=x?

Let's start by going back to some basics of solving single
variable equations. One of the fundamental tools we use in algebra is the idea of
inverse operations. Inverse means opposite, and operations means things like add,
subtract, divide, multiply, take a root, or raise to a power. Inverse operations are
operations that "undo" each other so to speak. Addition and subtraction undo each other;
multiplying and dividing undo each other; Raising to a power and taking a root undo each
other. Look at this simple example: x + 2 = 5

To solve this
problem we need to "undo" everything that is happening to x. If you examine the left
side of the equation, you will see that x is being added by 2. The inverse/opposite of
this would be subtracting 2, and that is what we will do to "undo" what has happened to
x. We are allowed to do this, or any other operation we choose, so long as we do the
same thing equally on both sides of the equal sign.

Think
of an equation like a balance scale. In a balance scale you have two bowls attached to a
lever of some kind, and the two bowls are balanced on opposite ends of the lever. If you
put 1lb in the left bowl, it will drop down lower than the right bowl. To "rebalance"
the scale, we need to put the same amount of weight (1lb) in the right bowl, and the two
bowls will again be balanced. Equations are like this. If we subtract 2 from one side of
the equal sign, then we must also subtract 2 from the other side of the equal sign to
maintain the balance of the equation.

x + 2 =
5

-2 -2

x =
3

So now on to your problem: square
root(2x+8)=x

As we did in the simple example of x + 2 = 5,
we want to undo any operations that have been done to x, so the first step is to remove
the radical (square root box) that has the x trapped inside. To do this, we need the
operation that is the opposite of taking a square root, which is raising to the second
power or squaring. So we are going to raise everything on both sides of the equal sign
to the second power. On the left side of the equal sign, what will happen is that the
square root and the squaring(raising to the second power) will cancel each other
out.

This is the same as the idea from the simpler example
of x + 2 = 5. When we subtracted to from the left side of that problem, the +2 and -2
cancelled each other out because they were inverse/opposite
operations.

Now back to our problem. We've raised the left
side to the second power, so remember our scale, we need to do the same thing on the
right side. After squaring both sides we should
have:

2x+8=x squared

Now we
have a quadratic equation which we can solve either by factoring or using the quadratic
formula. We'll factor this one.

First we have to get
everything on one side and set the problem equal to zero, which we can do by subtracting
2x and subtracting 8 from both sides. That will leave us 0=x squared -2x-8. We will
factor the quadratic term, x squared, which is x times x and those will become the first
factor in each of our binomial factors:

0=(x )(x
)

Now we must find the factors of our constant -8 that have
a sum/difference of -2. These are -4 and +2, because (-4)(2) is -8, but -4+2 is -2. We
now put those in our binomial
factors:

0=(x-4)(x+2)

Lastly,
we use the zero product property to find our solutions. Take the two factors and set
them each equal to zero and solve:

x-4=0 and
x+2=0

+4 +4 -2 -2

so x=4 or
x=-2, these are your two real solutions.