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Start with the given expression
Factor out the GCF
Now let's focus on the inner expression
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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
:
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to
.
1*36
2*18
3*12
4*9
6*6
(-1)*(-36)
(-2)*(-18)
(-3)*(-12)
(-4)*(-9)
(-6)*(-6)
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 | 36 | 1+36=37 |
| 2 | 18 | 2+18=20 |
| 3 | 12 | 3+12=15 |
| 4 | 9 | 4+9=13 |
| 6 | 6 | 6+6=12 |
| -1 | -36 | -1+(-36)=-37 |
| -2 | -18 | -2+(-18)=-20 |
| -3 | -12 | -3+(-12)=-15 |
| -4 | -9 | -4+(-9)=-13 |
| -6 | -6 | -6+(-6)=-12 |
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
Condense the common factors.
So the expression goes from
and factors further to
So
factors to
.
Set the factored expression equal to zero
Divide both sides by 5.
Take the square root of both sides
Add
to both sides.
Combine like terms on the right side.
Divide both sides by
to isolate
.
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Answer:
So the answer is
Which approximates to
