Answer:
The Completing the Square Method basically "forces" the existence of a perfect square trinomial in order to easily factor an equation where factoring by grouping is impossible.
For
x
2
−
8
x
−
15
, the first step would be to make the equation equal to zero and add 15 to both sides.
x
2
−
8
x
=
15
Next, we need to turn the binomial on the left side of the equation into a perfect trinomial. We can do this by dividing the coefficient of the "middle x-term", which would be
−
8
, in half then squaring it.
−
8
2
=
−
4
and
(
−
4
)
2
=
16
We then add the result to both sides of the equation.
x
2
−
8
x
+
16
=
15
+
16
Now, we can factor the perfect square trinomial and simplify
15
+
16
.
(
x
+
4
)
2
=
31
Now, we subtract
31
from both sides of the equation.
(
x
+
4
)
2
−
31
In order to factor these terms, both of them need to be the "squared version of their square rooted form" so that the terms stay the same when factoring.
(
x
+
4
)
2
−
√
31
2
Now, we can factor them. Since the expression follows the case
a
2
−
b
2
, where, in this case,
a
=
(
x
+
4
)
and
b
=
√
31
, we can factor them by following the Difference of Squares Formula:
a
2
−
b
2
=
(
a
+
b
)
(
a
−
b
)
.
Your final answer would be:
(
(
x
+
4
)
+
√
31
)
(
(
x
+
4
)
−
√
3