SOLUTION: Please solve this quadratic equation by completing the square and factoring: 2m^2 + 5m = 12

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Question 472964: Please solve this quadratic equation by completing the square and factoring:
2m^2 + 5m = 12
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Completing the square:

2m%5E2+%2B+5m+=+12 Start with the given equation.


2m%5E2+%2B+5m+-+12=0 Subtract 12 from both sides.



Now let's complete the square for the left side.


2m%5E2%2B5m-12 Start with the given expression.


2%28m%5E2%2B%285%2F2%29m-6%29 Factor out the m%5E2 coefficient 2. This step is very important: the m%5E2 coefficient must be equal to 1.


Take half of the m 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%28m%5E2%2B%285%2F2%29m%2Bhighlight%2825%2F16-25%2F16%29-6%29 Now add and subtract 25%2F16 inside the parenthesis. Make sure to place this after the "m" term. Notice how 25%2F16-25%2F16=0. So the expression is not changed.


2%28%28m%5E2%2B%285%2F2%29m%2B25%2F16%29-25%2F16-6%29 Group the first three terms.


2%28%28m%2B5%2F4%29%5E2-25%2F16-6%29 Factor m%5E2%2B%285%2F2%29m%2B25%2F16 to get %28m%2B5%2F4%29%5E2.


2%28%28m%2B5%2F4%29%5E2-121%2F16%29 Combine like terms.


2%28m%2B5%2F4%29%5E2%2B2%28-121%2F16%29 Distribute.


2%28m%2B5%2F4%29%5E2-121%2F8 Multiply.


So after completing the square, 2m%5E2%2B5m-12 transforms to 2%28m%2B5%2F4%29%5E2-121%2F8. So 2m%5E2%2B5m-12=2%28m%2B5%2F4%29%5E2-121%2F8.


So 2m%5E2%2B5m-12=0 is equivalent to 2%28m%2B5%2F4%29%5E2-121%2F8=0.


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Now let's solve 2%28m%2B5%2F4%29%5E2-121%2F8=0


2%28m%2B5%2F4%29%5E2-121%2F8=0 Start with the given equation.


2%28m%2B5%2F4%29%5E2=0%2B121%2F8 Add 121%2F8 to both sides.


2%28m%2B5%2F4%29%5E2=121%2F8 Combine like terms.


%28m%2B5%2F4%29%5E2=%28121%2F8%29%2F%282%29 Divide both sides by 2.


%28m%2B5%2F4%29%5E2=121%2F16 Reduce.


m%2B5%2F4=%22%22%2B-sqrt%28121%2F16%29 Take the square root of both sides.


m%2B5%2F4=sqrt%28121%2F16%29 or m%2B5%2F4=-sqrt%28121%2F16%29 Break up the "plus/minus" to form two equations.


m%2B5%2F4=11%2F4 or m%2B5%2F4=-11%2F4 Take the square root of 121%2F16 to get 11%2F4.


m=-5%2F4%2B11%2F4 or m=-5%2F4-11%2F4 Subtract 5%2F4 from both sides.


m=3%2F2 or m=-4 Combine like terms.


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Answer:


So the solutions are m=3%2F2 or m=-4.



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Factoring:

2m%5E2+%2B+5m+=+12 Start with the given equation.


2m%5E2+%2B+5m+-+12=0 Subtract 12 from both sides.


Now let's factor:





Looking at the expression 2m%5E2%2B5m-12, we can see that the first coefficient is 2, the second coefficient is 5, and the last term is -12.


Now multiply the first coefficient 2 by the last term -12 to get %282%29%28-12%29=-24.


Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient 5?


To find these two numbers, we need to list all of the factors of -24 (the previous product).


Factors of -24:
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -24.
1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:


First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From the table, we can see that the two numbers -3 and 8 add to 5 (the middle coefficient).


So the two numbers -3 and 8 both multiply to -24 and add to 5


Now replace the middle term 5m with -3m%2B8m. Remember, -3 and 8 add to 5. So this shows us that -3m%2B8m=5m.


2m%5E2%2Bhighlight%28-3m%2B8m%29-12 Replace the second term 5m with -3m%2B8m.


%282m%5E2-3m%29%2B%288m-12%29 Group the terms into two pairs.


m%282m-3%29%2B%288m-12%29 Factor out the GCF m from the first group.


m%282m-3%29%2B4%282m-3%29 Factor out 4 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28m%2B4%29%282m-3%29 Combine like terms. Or factor out the common term 2m-3


So 2m%5E2%2B5m-12 factors to %28m%2B4%29%282m-3%29.


In other words, 2m%5E2%2B5m-12=%28m%2B4%29%282m-3%29.



So 2m%5E2%2B5m-12=0 turns into %28m%2B4%29%282m-3%29=0


%28m%2B4%29%282m-3%29=0 Start with the given equation


m%2B4=0 or 2m-3=0 Use the zero product property


m=-4 or m=3%2F2 Solve for 'm' in each equation.


So the solutions are m=-4 or m=3%2F2 (which are the same as the ones above)