Precalculus Examples

$3 (2^{2 x}) - 11 \cdot 2^{x} - 4 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

$3 \cdot {(2^{x})}^{2} - 11 \cdot 2^{x} - 4 = 0$

Step 1.2

Let $u = 2^{x}$ . Substitute $u$ for all occurrences of $2^{x}$ .

$3 u^{2} - 11 u - 4 = 0$

Step 1.3

Factor by grouping.

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Step 1.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = - 11$ .

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Step 1.3.1.1

Factor $- 11$ out of $- 11 u$ .

$3 u^{2} - 11 u - 4 = 0$

Step 1.3.1.2

Rewrite $- 11$ as $1$ plus $- 12$

$3 u^{2} + (1 - 12) u - 4 = 0$

Step 1.3.1.3

Apply the distributive property.

$3 u^{2} + 1 u - 12 u - 4 = 0$

Step 1.3.1.4

Multiply $u$ by $1$ .

$3 u^{2} + u - 12 u - 4 = 0$

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

$(3 u^{2} + u) - 12 u - 4 = 0$

Step 1.3.2.2

Factor out the greatest common factor (GCF) from each group.

$u (3 u + 1) - 4 (3 u + 1) = 0$

Step 1.3.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$(3 u + 1) (u - 4) = 0$

Step 1.4

Replace all occurrences of $u$ with $2^{x}$ .

$(3 \cdot 2^{x} + 1) (2^{x} - 4) = 0$

Step 2

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

$3 \cdot 2^{x} + 1 = 0$

$2^{x} - 4 = 0$

Step 3

Set $3 \cdot 2^{x} + 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3 \cdot 2^{x} + 1$ equal to $0$ .

$3 \cdot 2^{x} + 1 = 0$

Step 3.2

Solve $3 \cdot 2^{x} + 1 = 0$ for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$3 \cdot 2^{x} = - 1$

Step 3.2.2

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

$\ln (3 \cdot 2^{x}) = \ln (- 1)$

Step 3.2.3

The equation cannot be solved because $\ln (- 1)$ is undefined.

Undefined

Step 3.2.4

There is no solution for $3 \cdot 2^{x} = - 1$

No solution

Step 4

Set $2^{x} - 4$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $2^{x} - 4$ equal to $0$ .

$2^{x} - 4 = 0$

Step 4.2

Solve $2^{x} - 4 = 0$ for $x$ .

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Step 4.2.1

Add $4$ to both sides of the equation.

$2^{x} = 4$

Step 4.2.2

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

$2^{x} = 2^{2}$

Step 4.2.3

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

$x = 2$

Step 5

The final solution is all the values that make $(3 \cdot 2^{x} + 1) (2^{x} - 4) = 0$ true.

$x = 2$