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Let’s check out your problem:
2
(
3
x
+
1
)
+
7
=
3
2
x
2(3^x+1) +7= 3^{2x}
2
(
3
x
+
1
)
+
7
=
3
2
x
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Home
Math Problems
Algebra 2
Factor by grouping
Full solution
Q.
2
(
3
x
+
1
)
+
7
=
3
2
x
2(3^x+1) +7= 3^{2x}
2
(
3
x
+
1
)
+
7
=
3
2
x
Simplify Equation:
First, let's simplify the equation by distributing the
2
2
2
into the parentheses:
2
×
3
x
+
2
×
1
+
7
=
3
(
2
x
)
2 \times 3^x + 2 \times 1 + 7 = 3^{(2x)}
2
×
3
x
+
2
×
1
+
7
=
3
(
2
x
)
.
\newline
So we get
2
×
3
x
+
2
+
7
=
3
(
2
x
)
2 \times 3^x + 2 + 7 = 3^{(2x)}
2
×
3
x
+
2
+
7
=
3
(
2
x
)
.
Combine Like Terms:
Now, combine like terms on the left side:
2
×
3
x
+
9
=
3
2
x
2 \times 3^x + 9 = 3^{2x}
2
×
3
x
+
9
=
3
2
x
.
Isolate Terms with x:
Subtract
9
9
9
from both sides to isolate terms with
x
x
x
on one side:
2
×
3
x
=
3
2
x
−
9
2 \times 3^x = 3^{2x} - 9
2
×
3
x
=
3
2
x
−
9
.
Write in Same Base:
We can write
3
2
x
3^{2x}
3
2
x
as
(
3
x
)
2
(3^x)^2
(
3
x
)
2
to have the same base:
2
×
3
x
=
(
3
x
)
2
−
9
2 \times 3^x = (3^x)^2 - 9
2
×
3
x
=
(
3
x
)
2
−
9
.
Set Equation to Zero:
Let's set this equation equal to zero:
(
3
x
)
2
−
2
⋅
3
x
−
9
=
0
(3^x)^2 - 2 \cdot 3^x - 9 = 0
(
3
x
)
2
−
2
⋅
3
x
−
9
=
0
.
Substitution Method:
Now, we can use a substitution method. Let's let
u
=
3
x
u = 3^x
u
=
3
x
. So our equation becomes
u
2
−
2
u
−
9
=
0
u^2 - 2u - 9 = 0
u
2
−
2
u
−
9
=
0
.
Factor Quadratic Equation:
We can factor this quadratic equation:
(
u
−
3
)
(
u
+
3
)
=
0
(u - 3)(u + 3) = 0
(
u
−
3
)
(
u
+
3
)
=
0
.
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\newline
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\newline
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