In order to factor , first we need to ask ourselves: What two numbers multiply to -36 and add to 0? Lets find out by listing all of the possible factors of -36
Factors:
1,2,3,4,6,9,12,18,36,
-1,-2,-3,-4,-6,-9,-12,-18,-36,List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to -36.
(-1)*(36)=-36
(-2)*(18)=-36
(-3)*(12)=-36
(-4)*(9)=-36
(-6)*(6)=-36
Now which of these pairs add to 0? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 0
First Number | | | Second Number | | | Sum | 1 | | | -36 | || | 1+(-36)=-35 | 2 | | | -18 | || | 2+(-18)=-16 | 3 | | | -12 | || | 3+(-12)=-9 | 4 | | | -9 | || | 4+(-9)=-5 | 6 | | | -6 | || | 6+(-6)=0 | -1 | | | 36 | || | (-1)+36=35 | -2 | | | 18 | || | (-2)+18=16 | -3 | | | 12 | || | (-3)+12=9 | -4 | | | 9 | || | (-4)+9=5 | -6 | | | 6 | || | (-6)+6=0 | We can see from the table that -6 and 6 add to 0.So the two numbers that multiply to -36 and add to 0 are: -6 and 6
Now we substitute these numbers into a and b of the general equation of a product of linear factors which is:
substitute a=-6 and b=6
So the equation becomes:
(x-6)(x+6)
Notice that if we foil (x-6)(x+6) we get the quadratic again
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Factor the first denominator
Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1) |
In order to factor , first we need to ask ourselves: What two numbers multiply to -4 and add to 0? Lets find out by listing all of the possible factors of -4
Factors:
1,2,4,4,6,9,12,18,36,
-1,-2,-4,-4,-6,-9,-12,-18,-36,List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to -4.
(-1)*(36)=-4
(-2)*(18)=-4
(-4)*(12)=-4
(-4)*(9)=-4
(-6)*(6)=-4
Now which of these pairs add to 0? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 0
First Number | | | Second Number | | | Sum | 1 | | | -36 | || | 1+(-36)=-35 | 2 | | | -18 | || | 2+(-18)=-16 | 4 | | | -12 | || | 4+(-12)=-8 | 4 | | | -9 | || | 4+(-9)=-5 | 6 | | | -6 | || | 6+(-6)=0 | -1 | | | 36 | || | (-1)+36=35 | -2 | | | 18 | || | (-2)+18=16 | -4 | | | 12 | || | (-4)+12=8 | -4 | | | 9 | || | (-4)+9=5 | -6 | | | 6 | || | (-6)+6=0 | We can see from the table that -6 and 6 add to 0.So the two numbers that multiply to -4 and add to 0 are: -6 and 6
Now we substitute these numbers into a and b of the general equation of a product of linear factors which is:
substitute a=-6 and b=6
So the equation becomes:
(x-6)(x+6)
Notice that if we foil (x-6)(x+6) we get the quadratic again
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Factor the second numerator
Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1) |
In order to factor , first we need to ask ourselves: What two numbers multiply to 8 and add to 6? Lets find out by listing all of the possible factors of 8
Factors:
1,2,4,8,6,9,12,18,36,
-1,-2,-4,-8,-6,-9,-12,-18,-36,List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to 8.
1*36=8
2*18=8
4*12=8
8*9=8
6*6=8
(-1)*(-36)=8
(-2)*(-18)=8
(-4)*(-12)=8
(-8)*(-9)=8
(-6)*(-6)=8
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6
First Number | | | Second Number | | | Sum | 1 | | | 36 | || | 1+36=37 | 2 | | | 18 | || | 2+18=20 | 4 | | | 12 | || | 4+12=16 | 8 | | | 9 | || | 8+9=17 | 6 | | | 6 | || | 6+6=12 | -1 | | | -36 | || | -1+(-36)=-37 | -2 | | | -18 | || | -2+(-18)=-20 | -4 | | | -12 | || | -4+(-12)=-16 | -8 | | | -9 | || | -8+(-9)=-17 | -6 | | | -6 | || | -6+(-6)=-12 | substitute a=-6 and b=6
So the equation becomes:
(x-6)(x+6)
Notice that if we foil (x-6)(x+6) we get the quadratic again
None of these factors add to 6. So this quadratic cannot be factored. In order to solve for x, we need to use the quadratic formula.
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Factor the second denominator
Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1) |
In order to factor , first we need to ask ourselves: What two numbers multiply to -24 and add to -2? Lets find out by listing all of the possible factors of -24
Factors:
1,2,3,4,6,8,12,24,36,
-1,-2,-3,-4,-6,-8,-12,-24,-36,List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to -24.
(-1)*(36)=-24
(-2)*(24)=-24
(-3)*(12)=-24
(-4)*(8)=-24
(-6)*(6)=-24
Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2
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