SOLUTION: Please help me solve this equation: 4x^2+6x-6=2 It is called completing the square and has to do with solving quadratic equations.

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Question 196866This question is from textbook McDougal Littell Algebra 1
: Please help me solve this equation: 4x^2+6x-6=2
It is called completing the square and has to do with solving quadratic equations. This question is from textbook McDougal Littell Algebra 1

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
4x%5E2%2B6x-6=2 Start with the given equation.


4x%5E2%2B6x-6-2=0 Subtract 2 from both sides.


4x%5E2%2B6x-8=0 Combine like terms.


Now let's complete the square for the expression on the left side:



4x%5E2%2B6x-8 Start with the given expression.


4%28x%5E2%2B%283%2F2%29x-2%29 Factor out the x%5E2 coefficient 4. This step is very important: the x%5E2 coefficient must be equal to 1.


Take half of the x coefficient 3%2F2 to get 3%2F4. In other words, %281%2F2%29%283%2F2%29=3%2F4.


Now square 3%2F4 to get 9%2F16. In other words, %283%2F4%29%5E2=%283%2F4%29%283%2F4%29=9%2F16


4%28x%5E2%2B%283%2F2%29x%2Bhighlight%289%2F16-9%2F16%29-2%29 Now add and subtract 9%2F16 inside the parenthesis. Make sure to place this after the "x" term. Notice how 9%2F16-9%2F16=0. So the expression is not changed.


4%28%28x%5E2%2B%283%2F2%29x%2B9%2F16%29-9%2F16-2%29 Group the first three terms.


4%28%28x%2B3%2F4%29%5E2-9%2F16-2%29 Factor x%5E2%2B%283%2F2%29x%2B9%2F16 to get %28x%2B3%2F4%29%5E2.


4%28%28x%2B3%2F4%29%5E2-41%2F16%29 Combine like terms.


4%28x%2B3%2F4%29%5E2%2B4%28-41%2F16%29 Distribute.


4%28x%2B3%2F4%29%5E2-41%2F4 Multiply.


So after completing the square, 4x%5E2%2B6x-8 transforms to 4%28x%2B3%2F4%29%5E2-41%2F4. So 4x%5E2%2B6x-8=4%28x%2B3%2F4%29%5E2-41%2F4.


So 4x%5E2%2B6x-8=0 is equivalent to 4%28x%2B3%2F4%29%5E2-41%2F4=0.


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Now let's solve 4%28x%2B3%2F4%29%5E2-41%2F4=0


4%28x%2B3%2F4%29%5E2-41%2F4=0 Start with the given equation.


4%28x%2B3%2F4%29%5E2=0%2B41%2F4Add 41%2F4 to both sides.


4%28x%2B3%2F4%29%5E2=41%2F4 Combine like terms.


%28x%2B3%2F4%29%5E2=%2841%2F4%29%2F%284%29 Divide both sides by 4.


%28x%2B3%2F4%29%5E2=41%2F16 Reduce.


x%2B3%2F4=%22%22%2B-sqrt%2841%2F16%29 Take the square root of both sides.


x%2B3%2F4=sqrt%2841%2F16%29 or x%2B3%2F4=-sqrt%2841%2F16%29 Break up the "plus/minus" to form two equations.


x%2B3%2F4=sqrt%2841%29%2F4 or x%2B3%2F4=-sqrt%2841%29%2F4 Simplify the square root.


x=-3%2F4%2Bsqrt%2841%29%2F4 or x=-3%2F4-sqrt%2841%29%2F4 Subtract 3%2F4 from both sides.


x=%28-3%2Bsqrt%2841%29%29%2F%284%29 or x=%28-3-sqrt%2841%29%29%2F%284%29 Combine the fractions.


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Answer:


So the solutions are x=%28-3%2Bsqrt%2841%29%29%2F%284%29 or x=%28-3-sqrt%2841%29%29%2F%284%29.