SOLUTION: Solve by completing the square. 2x^2+8x+5=0

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Question 299406: Solve by completing the square. 2x^2+8x+5=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

2x%5E2%2B8x%2B5 Start with the given expression.


2%28x%5E2%2B4x%2B5%2F2%29 Factor out the x%5E2 coefficient 2. 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%284%29=2.


Now square 2 to get 4. In other words, %282%29%5E2=%282%29%282%29=4


2%28x%5E2%2B4x%2Bhighlight%284-4%29%2B5%2F2%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.


2%28%28x%5E2%2B4x%2B4%29-4%2B5%2F2%29 Group the first three terms.


2%28%28x%2B2%29%5E2-4%2B5%2F2%29 Factor x%5E2%2B4x%2B4 to get %28x%2B2%29%5E2.


2%28%28x%2B2%29%5E2-3%2F2%29 Combine like terms.


2%28x%2B2%29%5E2%2B2%28-3%2F2%29 Distribute.


2%28x%2B2%29%5E2-3 Multiply.


So after completing the square, 2x%5E2%2B8x%2B5 transforms to 2%28x%2B2%29%5E2-3. So 2x%5E2%2B8x%2B5=2%28x%2B2%29%5E2-3.


So 2x%5E2%2B8x%2B5=0 is equivalent to 2%28x%2B2%29%5E2-3=0.


Now let's solve 2%28x%2B2%29%5E2-3=0



2%28x%2B2%29%5E2=0%2B3Add 3 to both sides.


2%28x%2B2%29%5E2=3 Combine like terms.


%28x%2B2%29%5E2=%283%29%2F%282%29 Divide both sides by 2.


x%2B2=%22%22%2B-sqrt%283%2F2%29 Take the square root of both sides.


x%2B2=sqrt%283%2F2%29 or x%2B2=-sqrt%283%2F2%29 Break up the "plus/minus" to form two equations.


x%2B2=sqrt%286%29%2F2 or x%2B2=-sqrt%286%29%2F2 Simplify the square root.


x=-2%2Bsqrt%286%29%2F2 or x=-2-sqrt%286%29%2F2 Subtract 2 from both sides.


x=%28-4%2Bsqrt%286%29%29%2F%282%29 or x=%28-4-sqrt%286%29%29%2F%282%29 Combine the fractions.


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Answer:


So the solutions are x=%28-4%2Bsqrt%286%29%29%2F%282%29 or x=%28-4-sqrt%286%29%29%2F%282%29.