SOLUTION: Complete the square,simplify. Express any complex numbers using i notation {{{2x^2+5x-9=0}}} (2x )(x ) I am stuck because I can not figure out which 2 numbers multiply t

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: Complete the square,simplify. Express any complex numbers using i notation {{{2x^2+5x-9=0}}} (2x )(x ) I am stuck because I can not figure out which 2 numbers multiply t      Log On


   

Question 264629: Complete the square,simplify. Express any complex numbers using i notation
2x%5E2%2B5x-9=0
(2x )(x )
I am stuck because I can not figure out which 2 numbers multiply to 9 and add to 5.If I use (2x+3)(x+1) I can get 5x but no 9. Please help! This is where I always get stuck while factoring.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
It turns out that you can't factor 2x%5E2%2B5x-9 over the rationals. So let's follow the instructions and complete the square.



2x%5E2%2B5x-9 Start with the given expression.


2%28x%5E2%2B%285%2F2%29x-9%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 5%2F2 to get 5%2F4. In other words, %281%2F2%29%285%2F2%29=5%2F4.


Now square 5%2F4 to get 25%2F16. In other words, %285%2F4%29%5E2=%285%2F4%29%285%2F4%29=25%2F16


2%28x%5E2%2B%285%2F2%29x%2Bhighlight%2825%2F16-25%2F16%29-9%2F2%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%2B%285%2F2%29x%2B25%2F16%29-25%2F16-9%2F2%29 Group the first three terms.


2%28%28x%2B5%2F4%29%5E2-25%2F16-9%2F2%29 Factor x%5E2%2B%285%2F2%29x%2B25%2F16 to get %28x%2B5%2F4%29%5E2.


2%28%28x%2B5%2F4%29%5E2-97%2F16%29 Combine like terms.


2%28x%2B5%2F4%29%5E2%2B2%28-97%2F16%29 Distribute.


2%28x%2B5%2F4%29%5E2-97%2F8 Multiply.


So after completing the square, 2x%5E2%2B5x-9 transforms to 2%28x%2B5%2F4%29%5E2-97%2F8. So 2x%5E2%2B5x-9=2%28x%2B5%2F4%29%5E2-97%2F8.


So 2x%5E2%2B5x-9=0 is equivalent to 2%28x%2B5%2F4%29%5E2-97%2F8=0.


Now let's solve 2%28x%2B5%2F4%29%5E2-97%2F8=0




2%28x%2B5%2F4%29%5E2-97%2F8=0 Start with the given equation.


2%28x%2B5%2F4%29%5E2=0%2B97%2F8Add 97%2F8 to both sides.


2%28x%2B5%2F4%29%5E2=97%2F8 Combine like terms.


%28x%2B5%2F4%29%5E2=%2897%2F8%29%2F%282%29 Divide both sides by 2.


%28x%2B5%2F4%29%5E2=97%2F16 Reduce.


x%2B5%2F4=%22%22%2B-sqrt%2897%2F16%29 Take the square root of both sides.


x%2B5%2F4=sqrt%2897%2F16%29 or x%2B5%2F4=-sqrt%2897%2F16%29 Break up the "plus/minus" to form two equations.


x%2B5%2F4=sqrt%2897%29%2F4 or x%2B5%2F4=-sqrt%2897%29%2F4 Simplify the square root.


x=-5%2F4%2Bsqrt%2897%29%2F4 or x=-5%2F4-sqrt%2897%29%2F4 Subtract 5%2F4 from both sides.


x=%28-5%2Bsqrt%2897%29%29%2F%284%29 or x=%28-5-sqrt%2897%29%29%2F%284%29 Combine the fractions.


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Answer:


So the solutions are x=%28-5%2Bsqrt%2897%29%29%2F%284%29 or x=%28-5-sqrt%2897%29%29%2F%284%29.