Finite Math Examples

$2^{2 x} + 128 = 24 \cdot 2^{x}$

Step 1

Subtract $24 \cdot 2^{x}$ from both sides of the equation.

$2^{2 x} + 128 - 24 \cdot 2^{x} = 0$

Step 2

Rewrite $2^{2 x}$ as exponentiation.

${(2^{x})}^{2} + 128 - 24 \cdot 2^{x} = 0$

Step 3

Substitute $u$ for $2^{x}$ .

$u^{2} + 128 - 24 u = 0$

Step 4

Move $128$ .

$u^{2} - 24 u + 128 = 0$

Step 5

Solve for $u$ .

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Step 5.1

Factor $u^{2} - 24 u + 128$ using the AC method.

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Step 5.1.1

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 $128$ and whose sum is $- 24$ .

$- 16, - 8$

Step 5.1.2

Write the factored form using these integers.

$(u - 16) (u - 8) = 0$

Step 5.2

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

$u - 16 = 0$

$u - 8 = 0$

Step 5.3

Set $u - 16$ equal to $0$ and solve for $u$ .

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Step 5.3.1

Set $u - 16$ equal to $0$ .

$u - 16 = 0$

Step 5.3.2

Add $16$ to both sides of the equation.

$u = 16$

Step 5.4

Set $u - 8$ equal to $0$ and solve for $u$ .

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Step 5.4.1

Set $u - 8$ equal to $0$ .

$u - 8 = 0$

Step 5.4.2

Add $8$ to both sides of the equation.

$u = 8$

Step 5.5

The final solution is all the values that make $(u - 16) (u - 8) = 0$ true.

$u = 16, 8$

Step 6

Substitute $16$ for $u$ in $u = 2^{x}$ .

$16 = 2^{x}$

Step 7

Solve $16 = 2^{x}$ .

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Step 7.1

Rewrite the equation as $2^{x} = 16$ .

$2^{x} = 16$

Step 7.2

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

$2^{x} = 2^{4}$

Step 7.3

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

$x = 4$

Step 8

Substitute $8$ for $u$ in $u = 2^{x}$ .

$8 = 2^{x}$

Step 9

Solve $8 = 2^{x}$ .

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Step 9.1

Rewrite the equation as $2^{x} = 8$ .

$2^{x} = 8$

Step 9.2

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

$2^{x} = 2^{3}$

Step 9.3

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

$x = 3$

Step 10

List the solutions that makes the equation true.

$x = 4, 3$