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.

Algebra ->  Equations -> 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.      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
a)
5%2Asqrt%28x%29-x=6

5%2Asqrt%28x%29=6%2Bx Add x to both sides


%285%2Asqrt%28x%29%29%5E2=%286%2Bx%29%5E2Square both sides

25x=%286%2Bx%29%5E2

25x=36%2B12x%2Bx%5E2 Foil

0=36%2B12x%2Bx%5E2-25x Subtract 25x from both sides

0=x%5E2-13x%2B36 Combine like terms and rearrange


Now lets use the quadratic formula to solve for x:

Starting with the general quadratic

ax%5E2%2Bbx%2Bc

the general form of the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve x%5E2-13%2Ax%2B36

x+=+%2813+%2B-+sqrt%28+%28-13%29%5E2-4%2A1%2A36+%29%29%2F%282%2A1%29 Plug in a=1, b=-13, and c=36



x+=+%2813+%2B-+sqrt%28+169-4%2A1%2A36+%29%29%2F%282%2A1%29 Square -13 to get 169



x+=+%2813+%2B-+sqrt%28+169%2B-144+%29%29%2F%282%2A1%29 Multiply -4%2A36%2A1 to get -144



x+=+%2813+%2B-+sqrt%28+25+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%2813+%2B-+5%29%2F%282%2A1%29 Simplify the square root



x+=+%2813+%2B-+5%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%2813+%2B+5%29%2F2 or x+=+%2813+-+5%29%2F2

Lets look at the first part:

x=18%2F2 Add the terms in the numerator
x=9 Divide

So one answer is
x=9
Now lets look at the second part:

x=8%2F2 Subtract the terms in the numerator
x=4 Divide

So another answer is
x=4

So our possible solutions are:
x=9 or x=4

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

Test the first solution

5%2Asqrt%289%29-9=6 Plug in x=9

5%2A3-9=6 Take the square root

15-9=6 Multiply

6=6 Combine like terms. Sot the solution works.

Now lets test the other solution

5%2Asqrt%284%29-4=6 Plug in x=4

5%2A2-4=6 Take the square root

10-4=6 Multiply

6=6 Combine like terms. Sot the solution works.

So our solutions are
x=4 or x=9


Here is the graph of y=5%2Asqrt%28x%29-x-6 (just get all terms to one side)

+graph%28+500%2C+500%2C+0%2C+15%2C+-5%2C+5%2C+5%2Asqrt%28x%29-x-6%29+ graph of y=5%2Asqrt%28x%29-x-6 where the roots represent the solutions we found

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b)
x%5E2-8=17

x%5E2-8-17=0Subtract 17 from both sides

Now lets use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc

the general form of the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve x%5E2-8%2Ax-17

x+=+%288+%2B-+sqrt%28+%28-8%29%5E2-4%2A1%2A-17+%29%29%2F%282%2A1%29 Plug in a=1, b=-8, and c=-17



x+=+%288+%2B-+sqrt%28+64-4%2A1%2A-17+%29%29%2F%282%2A1%29 Square -8 to get 64



x+=+%288+%2B-+sqrt%28+64%2B68+%29%29%2F%282%2A1%29 Multiply -4%2A-17%2A1 to get 68



x+=+%288+%2B-+sqrt%28+132+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%288+%2B-+2%2Asqrt%2833%29%29%2F%282%2A1%29 Simplify the square root



x+=+%288+%2B-+2%2Asqrt%2833%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%288+%2B+2%2Asqrt%2833%29%29%2F2 or x+=+%288+-+2%2Asqrt%2833%29%29%2F2


Which approximate to

x=9.74456264653803 or x=-1.74456264653803


So our solutions are:
x=9.74456264653803 or x=-1.74456264653803

Notice when we graph x%5E2-8%2Ax-17 we get:



when we use the root finder feature on our calculator, we find that x=9.74456264653803 and x=-1.74456264653803.So this verifies our answer