SOLUTION: Find all real or imaginary solutions to each equation. Us e the method of your choice. Sq rt of 7x+29 = x + 3

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Question 98541: Find all real or imaginary solutions to each equation. Us e the method of your choice.
Sq rt of 7x+29 = x + 3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%287x%2B29%29+=+x+%2B+3+ Start with the given equation

7x%2B29+=+%28x+%2B+3%29%5E2+ Square both sides


7x%2B29+=+x%5E2%2B6x%2B9+ Foil


0+=+x%5E2-x-20+ Get all terms to one side


Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using 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-x-20=0 ( notice a=1, b=-1, and c=-20)

x+=+%28--1+%2B-+sqrt%28+%28-1%29%5E2-4%2A1%2A-20+%29%29%2F%282%2A1%29 Plug in a=1, b=-1, and c=-20



x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%2A1%2A-20+%29%29%2F%282%2A1%29 Negate -1 to get 1



x+=+%281+%2B-+sqrt%28+1-4%2A1%2A-20+%29%29%2F%282%2A1%29 Square -1 to get 1 (note: remember when you square -1, you must square the negative as well. This is because %28-1%29%5E2=-1%2A-1=1.)



x+=+%281+%2B-+sqrt%28+1%2B80+%29%29%2F%282%2A1%29 Multiply -4%2A-20%2A1 to get 80



x+=+%281+%2B-+sqrt%28+81+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%281+%2B-+9%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%281+%2B-+9%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%281+%2B+9%29%2F2 or x+=+%281+-+9%29%2F2

Lets look at the first part:

x=%281+%2B+9%29%2F2

x=10%2F2 Add the terms in the numerator
x=5 Divide

So one answer is
x=5



Now lets look at the second part:

x=%281+-+9%29%2F2

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

So another answer is
x=-4

So our possible solutions are:
x=5 or x=-4


Check:
Lets check the possible solution x=5
sqrt%287x%2B29%29+=+x+%2B+3+ Start with the given equation

sqrt%287%285%29%2B29%29+=+5+%2B+3+ Plug in x=5

sqrt%2835%2B29%29+=+5+%2B+3+ Multiply

sqrt%2864%29+=+8+ Add

8+=+8+ Take the square root of both sides


Since the equation is true, x=5 is a solution

------------------------------
Lets check the possible solution x=-4
sqrt%287x%2B29%29+=+x+%2B+3+ Start with the given equation

sqrt%287%28-4%29%2B29%29+=+-4+%2B+3+ Plug in x=-4

sqrt%28-28%2B29%29+=+-4+%2B+3+ Multiply

sqrt%281%29+=+-1+ Add

1+=+-1+ Take the square root of both sides

Since the equation is not true, x=-4 is not a solution


So our only solution is x=5