SOLUTION: Hello. Use completing the square to solve each equation: -2 = 2x^2 - 5x thanks!

Algebra ->  Equations -> SOLUTION: Hello. Use completing the square to solve each equation: -2 = 2x^2 - 5x thanks!      Log On


   

Question 189934: Hello.
Use completing the square to solve each equation:
-2 = 2x^2 - 5x
thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
-2+=+2x%5E2+-+5x+ Start with the given equation.


0+=+2x%5E2+-+5x+%2B2+ Add 2 to both sides.


2x%5E2+-+5x+%2B2+=+0 Rearrange the equation


2%28x%5E2-%285%2F2%29x%2B1%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 -5%2F2 to get -5%2F4. In other words, %281%2F2%29%28-5%2F2%29=-5%2F4.


Now square -5%2F4 to get 25%2F16. In other words, %28-5%2F4%29%5E2=%28-5%2F4%29%28-5%2F4%29=25%2F16


2%28x%5E2-%285%2F2%29x%2Bhighlight%2825%2F16-25%2F16%29%2B1%29 Now add and subtract 25%2F16 inside the parenthesis. Make sure to place this after the "x" term. Notice how 25%2F16-25%2F16=0. So the expression is not changed.


2%28%28x%5E2-%285%2F2%29x%2B25%2F16%29-25%2F16%2B1%29 Group the first three terms.


2%28%28x-5%2F4%29%5E2-25%2F16%2B1%29 Factor x%5E2-%285%2F2%29x%2B25%2F16 to get %28x-5%2F4%29%5E2.


2%28%28x-5%2F4%29%5E2-9%2F16%29 Combine like terms.


2%28x-5%2F4%29%5E2%2B2%28-9%2F16%29 Distribute.


2%28x-5%2F4%29%5E2-9%2F8 Multiply.


So after completing the square, 2x%5E2-5x%2B2 transforms to 2%28x-5%2F4%29%5E2-9%2F8. So 2x%5E2-5x%2B2=2%28x-5%2F4%29%5E2-9%2F8.


So 2x%5E2-5x%2B2=0 is equivalent to 2%28x-5%2F4%29%5E2-9%2F8=0.



-------------------------------------------------------------------


2%28x-5%2F4%29%5E2-9%2F8=0 Start with the given equation.


2%28x-5%2F4%29%5E2=0%2B9%2F8Add 9%2F8 to both sides.


2%28x-5%2F4%29%5E2=9%2F8 Combine like terms.


%28x-5%2F4%29%5E2=%289%2F8%29%2F%282%29 Divide both sides by 2.


%28x-5%2F4%29%5E2=9%2F16 Reduce.


x-5%2F4=%22%22%2B-sqrt%289%2F16%29 Take the square root of both sides.


x-5%2F4=sqrt%289%2F16%29 or x-5%2F4=-sqrt%289%2F16%29 Break up the "plus/minus" to form two equations.


x-5%2F4=3%2F4 or x-5%2F4=-3%2F4 Take the square root of 9%2F16 to get 3%2F4.


x=5%2F4%2B3%2F4 or x=5%2F4-3%2F4 Add 5%2F4 to both sides.


x=2 or x=1%2F2 Combine like terms.


--------------------------------------


Answer:


So the solutions are x=2 or x=1%2F2.