Find All Complex Solutions 2^(2x)-12*2^x+2^5=0

$2^{2 x} - 12 \cdot 2^{x} + 2^{5} = 0$

Factor the left side of the equation.

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

${(2^{x})}^{2} - 12 \cdot 2^{x} + 2^{5} = 0$

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

$u^{2} - 12 u + 32 = 0$

Factor $u^{2} - 12 u + 32$ using the AC method.

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 $32$ and whose sum is $- 12$ .

$- 8, - 4$

Write the factored form using these integers.

$(u - 8) (u - 4) = 0$

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

$(2^{x} - 8) (2^{x} - 4) = 0$

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

$2^{x} - 8 = 0$

$2^{x} - 4 = 0$

Set $2^{x} - 8$ equal to $0$ and solve for $x$ .

Set $2^{x} - 8$ equal to $0$ .

$2^{x} - 8 = 0$

Solve $2^{x} - 8 = 0$ for $x$ .

Add $8$ to both sides of the equation.

$2^{x} = 8$

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

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

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

$x = 3$

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

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

$2^{x} - 4 = 0$

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

Add $4$ to both sides of the equation.

$2^{x} = 4$

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

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

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

$x = 2$

The final solution is all the values that make $(2^{x} - 8) (2^{x} - 4) = 0$ true.

$x = 3, 2$

Find All Complex Solutions 2^(2x)-12*2^x+2^5=0

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