Calculus Examples

$3^{2 x} - 108 \cdot 3^{x} + 2187 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $3^{2 x}$ as ${(3^{x})}^{2}$ .

${(3^{x})}^{2} - 108 \cdot 3^{x} + 2187 = 0$

Step 1.2

Let $u = 3^{x}$ . Substitute $u$ for all occurrences of $3^{x}$ .

$u^{2} - 108 u + 2187 = 0$

Step 1.3

Factor $u^{2} - 108 u + 2187$ using the AC method.

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Step 1.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2187$ and whose sum is $- 108$ .

$- 81, - 27$

Step 1.3.2

Write the factored form using these integers.

$(u - 81) (u - 27) = 0$

Step 1.4

Replace all occurrences of $u$ with $3^{x}$ .

$(3^{x} - 81) (3^{x} - 27) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3^{x} - 81 = 0$

$3^{x} - 27 = 0$

Step 3

Set $3^{x} - 81$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3^{x} - 81$ equal to $0$ .

$3^{x} - 81 = 0$

Step 3.2

Solve $3^{x} - 81 = 0$ for $x$ .

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Step 3.2.1

Add $81$ to both sides of the equation.

$3^{x} = 81$

Step 3.2.2

Create equivalent expressions in the equation that all have equal bases.

$3^{x} = 3^{4}$

Step 3.2.3

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x = 4$

Step 4

Set $3^{x} - 27$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $3^{x} - 27$ equal to $0$ .

$3^{x} - 27 = 0$

Step 4.2

Solve $3^{x} - 27 = 0$ for $x$ .

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Step 4.2.1

Add $27$ to both sides of the equation.

$3^{x} = 27$

Step 4.2.2

Create equivalent expressions in the equation that all have equal bases.

$3^{x} = 3^{3}$

Step 4.2.3

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x = 3$

Step 5

The final solution is all the values that make $(3^{x} - 81) (3^{x} - 27) = 0$ true.

$x = 4, 3$