Precalculus Examples

$9^{x} - 3^{x} - 6 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $9^{x}$ as ${(3^{2})}^{x}$ .

${(3^{2})}^{x} - 3^{x} - 6 = 0$

Step 1.2

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

${(3^{x})}^{2} - 3^{x} - 6 = 0$

Step 1.3

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

$u^{2} - u - 6 = 0$

Step 1.4

Factor $u^{2} - u - 6$ using the AC method.

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Step 1.4.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 1.4.2

Write the factored form using these integers.

$(u - 3) (u + 2) = 0$

Step 1.5

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

$(3^{x} - 3) (3^{x} + 2) = 0$

Step 2

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

$3^{x} - 3 = 0$

$3^{x} + 2 = 0$

Step 3

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

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Step 3.1

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

$3^{x} - 3 = 0$

Step 3.2

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

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Step 3.2.1

Add $3$ to both sides of the equation.

$3^{x} = 3$

Step 3.2.2

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

$3^{x} = 3^{1}$

Step 3.2.3

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

$x = 1$

Step 4

Set $3^{x} + 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $3^{x} + 2$ equal to $0$ .

$3^{x} + 2 = 0$

Step 4.2

Solve $3^{x} + 2 = 0$ for $x$ .

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Step 4.2.1

Subtract $2$ from both sides of the equation.

$3^{x} = - 2$

Step 4.2.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (3^{x}) = \ln (- 2)$

Step 4.2.3

The equation cannot be solved because $\ln (- 2)$ is undefined.

Undefined

Step 4.2.4

There is no solution for $3^{x} = - 2$

No solution

Step 5

The final solution is all the values that make $(3^{x} - 3) (3^{x} + 2) = 0$ true.

$x = 1$