SOLUTION: Solve by completing the square. 4x + 2 = x^2

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Question 156693This question is from textbook Algebra 1 An Incremental Development
: Solve by completing the square.
4x + 2 = x^2This question is from textbook Algebra 1 An Incremental Development

Answer by jim_thompson5910(28536) About Me  (Show Source):
You can put this solution on YOUR website!
4x+%2B+2+=+x%5E2 Start with the given equation


4x+%2B+2+-+x%5E2=0 Subtract x%5E2 from both sides


-+x%5E2%2B4x+%2B+2+=0 Rearrange the terms.

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So let's complete the square for -x%5E2%2B4x%2B2


-x%5E2%2B4x%2B2 Start with the given expression.


-1%28x%5E2-4x-2%29 Factor out the x%5E2 coefficient -1. This step is very important: the x%5E2 coefficient MUST be equal to 1.


Take half of the x coefficient -4 to get -2. In other words, %281%2F2%29%28-4%29=-2.


Now square -2 to get 4. In other words, %28-2%29%5E2=%28-2%29%28-2%29=4


-1%28x%5E2-4x%2Bhighlight%284-4%29-2%29 Now add and subtract 4 inside the parenthesis. Make sure to place this after the "x" term. Notice how 4-4=0. So the expression is not changed.


-1%28%28x%5E2-4x%2B4%29-4-2%29 Group the first three terms.


-1%28%28x-2%29%5E2-4-2%29 Factor x%5E2-4x%2B4 to get %28x-2%29%5E2.


-1%28%28x-2%29%5E2-6%29 Combine like terms.


-1%28x-2%29%5E2-1%28-6%29 Distribute.


-1%28x-2%29%5E2%2B6 Multiply.


So after completing the square, -x%5E2%2B4x%2B2 transforms to -1%28x-2%29%5E2%2B6. So -x%5E2%2B4x%2B2=-1%28x-2%29%5E2%2B6.


So -x%5E2%2B4x%2B2=0 is equivalent to -1%28x-2%29%5E2%2B6=0.


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-1%28x-2%29%5E2%2B6=0 Start with the given equation.


-1%28x-2%29%5E2=0-6Subtract 6 from both sides.


-1%28x-2%29%5E2=-6 Combine like terms.


%28x-2%29%5E2=%28-6%29%2F%28-1%29 Divide both sides by -1.


%28x-2%29%5E2=6 Reduce.


x-2=0%2B-sqrt%286%29 Take the square root of both sides. Note: remember the "plus/minus"


x-2=sqrt%286%29 or x-2=-sqrt%286%29 Break up the "plus/minus" to form two equations.


x=2%2Bsqrt%286%29 or x=2-sqrt%286%29 Add 2 to both sides.


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Answer:


So the solutions are x=2%2Bsqrt%286%29 or x=2-sqrt%286%29.