SOLUTION: Can you please help me with these equations? a) Find the roots of 5 sq.rt. of x-x=6 b) Find the roots of x^2-8=17. Thank you so much for your help.

Question 86200: Can you please help me with these equations?
a) Find the roots of 5 sq.rt. of x-x=6
b) Find the roots of x^2-8=17.
Thank you so much for your help.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)

Add x to both sides

Square both sides

Foil

Subtract from both sides

Combine like terms and rearrange

Now lets use the quadratic formula to solve for x:

Starting with the general quadratic

the general form of the quadratic equation is:

So lets solve

Plug in a=1, b=-13, and c=36

Square -13 to get 169

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root

Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

or

Lets look at the first part:

Add the terms in the numerator
Divide

So one answer is

Now lets look at the second part:

Subtract the terms in the numerator
Divide

So another answer is

So our possible solutions are:
or

Now we must check to see if the solutions work:
Check:

Test the first solution

Plug in x=9

Take the square root

Multiply

Combine like terms. Sot the solution works.

Now lets test the other solution

Plug in x=4

Take the square root

Multiply

Combine like terms. Sot the solution works.

So our solutions are
or

Here is the graph of (just get all terms to one side)

graph of where the roots represent the solutions we found

------------------------------------------------------------------------
b)

Subtract 17 from both sides

Now lets use the quadratic formula to solve for x:

Starting with the general quadratic

the general form of the quadratic equation is:

So lets solve

Plug in a=1, b=-8, and c=-17

Square -8 to get 64

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root

Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

or

Which approximate to

or

So our solutions are:
or

Notice when we graph we get:

when we use the root finder feature on our calculator, we find that and .So this verifies our answer
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