Finite Math Examples

$4^{x} + 7 \cdot 2^{x} = 44$

Step 1

Subtract $44$ from both sides of the equation.

$4^{x} + 7 \cdot 2^{x} - 44 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Rewrite $4^{x}$ as ${(2^{2})}^{x}$ .

${(2^{2})}^{x} + 7 \cdot 2^{x} - 44 = 0$

Step 2.2

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

${(2^{x})}^{2} + 7 \cdot 2^{x} - 44 = 0$

Step 2.3

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

$u^{2} + 7 u - 44 = 0$

Step 2.4

Factor $u^{2} + 7 u - 44$ using the AC method.

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Step 2.4.1

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 $- 44$ and whose sum is $7$ .

$- 4, 11$

Step 2.4.2

Write the factored form using these integers.

$(u - 4) (u + 11) = 0$

Step 2.5

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

$(2^{x} - 4) (2^{x} + 11) = 0$

Step 3

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

$2^{x} - 4 = 0$

$2^{x} + 11 = 0$

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

Set $2^{x} + 11$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2^{x} + 11$ equal to $0$ .

$2^{x} + 11 = 0$

Step 5.2

Solve $2^{x} + 11 = 0$ for $x$ .

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Step 5.2.1

Subtract $11$ from both sides of the equation.

$2^{x} = - 11$

Step 5.2.2

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

$\ln (2^{x}) = \ln (- 11)$

Step 5.2.3

The equation cannot be solved because $\ln (- 11)$ is undefined.

Undefined

Step 5.2.4

There is no solution for $2^{x} = - 11$

No solution

Step 6

The final solution is all the values that make $(2^{x} - 4) (2^{x} + 11) = 0$ true.

$x = 2$