Algebra Examples

$e^{2 x} - 4 e^{x} + 3 = 0$

Step 1

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

${(e^{x})}^{2} - 4 e^{x} + 3 = 0$

Step 2

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

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

Step 3

Factor $u^{2} - 4 u + 3$ using the AC method.

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Step 3.1

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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 3.2

Write the factored form using these integers.

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

Step 4

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

$(e^{x} - 3) (e^{x} - 1) = 0$