SOLUTION: How do I solve the follwoing system of equations? y = {{{ sqrt (x) }}}+ 4 y = 2x + 1 Can you provide a detailed answer showing all of the steps?

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Question 188800: How do I solve the follwoing system of equations?
y = +sqrt+%28x%29++ 4
y = 2x + 1


Can you provide a detailed answer showing all of the steps?
Found 2 solutions by jim_thompson5910, Alan3354:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
y+=++sqrt+%28x%29+%2B+4 Start with the first equation.


+2x+%2B+1+=++sqrt+%28x%29+%2B+4 Plug in y=2x%2B1


+2x+%2B+1+-4=++sqrt+%28x%29+ Subtract 4 from both sides.


+2x+-+3=++sqrt+%28x%29+ Combine like terms.


+%282x+-+3%29%5E2=++x+ Square both sides


+4x%5E2-12x%2B9=++x+ FOIL


+4x%5E2-12x%2B9-x=+0+ Subtract x from both sides.


+4x%5E2-13x%2B9=+0+ Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=4, b=-13, and c=9


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-13%29+%2B-+sqrt%28+%28-13%29%5E2-4%284%29%289%29+%29%29%2F%282%284%29%29 Plug in a=4, b=-13, and c=9


x+=+%2813+%2B-+sqrt%28+%28-13%29%5E2-4%284%29%289%29+%29%29%2F%282%284%29%29 Negate -13 to get 13.


x+=+%2813+%2B-+sqrt%28+169-4%284%29%289%29+%29%29%2F%282%284%29%29 Square -13 to get 169.


x+=+%2813+%2B-+sqrt%28+169-144+%29%29%2F%282%284%29%29 Multiply 4%284%29%289%29 to get 144


x+=+%2813+%2B-+sqrt%28+25+%29%29%2F%282%284%29%29 Subtract 144 from 169 to get 25


x+=+%2813+%2B-+sqrt%28+25+%29%29%2F%288%29 Multiply 2 and 4 to get 8.


x+=+%2813+%2B-+5%29%2F%288%29 Take the square root of 25 to get 5.


x+=+%2813+%2B+5%29%2F%288%29 or x+=+%2813+-+5%29%2F%288%29 Break up the expression.


x+=+%2818%29%2F%288%29 or x+=++%288%29%2F%288%29 Combine like terms.


x+=+9%2F4 or x+=+1 Simplify.


So the possible answers for "x" are x+=+9%2F4 or x+=+1


However, if you plug in x+=+1 into the equation +2x+%2B+1+=++sqrt+%28x%29+%2B+4, you'll find a contradiction. So this is an extraneous solution (ie we must ignore this value).


So the only solution for "x" is x+=+9%2F4


Now plug in x+=+9%2F4 into y=2x%2B1 to find "y"



Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
How do I solve the follwoing system of equations?
y =+sqrt+%28x%29+ + 4
y = 2x + 1
------------
There's more than one way, but since both right sides = y, I'd set them equal to each other.
2x+1 = sqrt(x) + 4
sqrt(x) = 2x-3
x = 4x^2 - 12x + 9
4x^2 - 13x - 9 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B-13x%2B-9+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-13%29%5E2-4%2A4%2A-9=313.

Discriminant d=313 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--13%2B-sqrt%28+313+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-13%29%2Bsqrt%28+313+%29%29%2F2%5C4+=+3.83647575161927
x%5B2%5D+=+%28-%28-13%29-sqrt%28+313+%29%29%2F2%5C4+=+-0.586475751619266

Quadratic expression 4x%5E2%2B-13x%2B-9 can be factored:
4x%5E2%2B-13x%2B-9+=+%28x-3.83647575161927%29%2A%28x--0.586475751619266%29
Again, the answer is: 3.83647575161927, -0.586475751619266. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-13%2Ax%2B-9+%29