Finite Math Examples

$49^{x} - 6 \cdot 7^{x} + 5 = 0$

Step 1

Rewrite $49^{x}$ as ${(7^{2})}^{x}$ .

${(7^{2})}^{x} - 6 \cdot 7^{x} + 5 = 0$

Step 2

Rewrite ${(7^{2})}^{x}$ as ${(7^{x})}^{2}$ .

${(7^{x})}^{2} - 6 \cdot 7^{x} + 5 = 0$

Step 3

Let $u = 7^{x}$ . Substitute $u$ for all occurrences of $7^{x}$ .

$u^{2} - 6 u + 5 = 0$

Step 4

Factor $u^{2} - 6 u + 5$ using the AC method.

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Step 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 $5$ and whose sum is $- 6$ .

$- 5, - 1$

Step 4.2

Write the factored form using these integers.

$(u - 5) (u - 1) = 0$

Step 5

Replace all occurrences of $u$ with $7^{x}$ .

$(7^{x} - 5) (7^{x} - 1) = 0$