Question 472964: Please solve this quadratic equation by completing the square and factoring:
2m^2 + 5m = 12
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website! Completing the square:
Start with the given equation.
Subtract 12 from both sides.
Now let's complete the square for the left side.
Start with the given expression.
Factor out the coefficient . This step is very important: the coefficient must be equal to 1.
Take half of the coefficient to get . In other words, .
Now square to get . In other words,
Now add and subtract inside the parenthesis. Make sure to place this after the "m" term. Notice how . So the expression is not changed.
Group the first three terms.
Factor to get .
Combine like terms.
Distribute.
Multiply.
So after completing the square, transforms to . So .
So is equivalent to .
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Now let's solve
Start with the given equation.
Add to both sides.
Combine like terms.
Divide both sides by .
Reduce.
Take the square root of both sides.
or Break up the "plus/minus" to form two equations.
or Take the square root of to get .
or Subtract from both sides.
or Combine like terms.
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Answer:
So the solutions are or .
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Factoring:
Start with the given equation.
Subtract 12 from both sides.
Now let's factor:
Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .
Now multiply the first coefficient by the last term to get .
Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
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 .
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 :
First Number | Second Number | Sum | 1 | -24 | 1+(-24)=-23 | 2 | -12 | 2+(-12)=-10 | 3 | -8 | 3+(-8)=-5 | 4 | -6 | 4+(-6)=-2 | -1 | 24 | -1+24=23 | -2 | 12 | -2+12=10 | -3 | 8 | -3+8=5 | -4 | 6 | -4+6=2 |
From the table, we can see that the two numbers and add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term with . Remember, and add to . So this shows us that .
Replace the second term with .
Group the terms into two pairs.
Factor out the GCF from the first group.
Factor out 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.
Combine like terms. Or factor out the common term
So factors to .
In other words, .
So turns into
Start with the given equation
or Use the zero product property
or Solve for 'm' in each equation.
So the solutions are or (which are the same as the ones above)
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