Finite Math Examples

$2 x^{4} - 166 x^{2} = - 324$

Step 1

Add $324$ to both sides of the equation.

$2 x^{4} - 166 x^{2} + 324 = 0$

Step 2

Factor $2$ out of $2 x^{4} - 166 x^{2} + 324$ .

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Step 2.1

Factor $2$ out of $2 x^{4}$ .

$2 (x^{4}) - 166 x^{2} + 324 = 0$

Step 2.2

Factor $2$ out of $- 166 x^{2}$ .

$2 (x^{4}) + 2 (- 83 x^{2}) + 324 = 0$

Step 2.3

Factor $2$ out of $324$ .

$2 x^{4} + 2 (- 83 x^{2}) + 2 \cdot 162 = 0$

Step 2.4

Factor $2$ out of $2 x^{4} + 2 (- 83 x^{2})$ .

$2 (x^{4} - 83 x^{2}) + 2 \cdot 162 = 0$

Step 2.5

Factor $2$ out of $2 (x^{4} - 83 x^{2}) + 2 \cdot 162$ .

$2 (x^{4} - 83 x^{2} + 162) = 0$

Step 3

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

$2 ({(x^{2})}^{2} - 83 x^{2} + 162) = 0$

Step 4

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

$2 (u^{2} - 83 u + 162) = 0$

Step 5

Factor $u^{2} - 83 u + 162$ using the AC method.

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Step 5.1

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 $162$ and whose sum is $- 83$ .

$- 81, - 2$

Step 5.2

Write the factored form using these integers.

$2 ((u - 81) (u - 2)) = 0$

Step 6

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

$2 ((x^{2} - 81) (x^{2} - 2)) = 0$

Step 7

Rewrite $81$ as $9^{2}$ .

$2 ((x^{2} - 9^{2}) (x^{2} - 2)) = 0$

Step 8

Factor.

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Step 8.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 9$ .

$2 ((x + 9) (x - 9) (x^{2} - 2)) = 0$

Step 8.2

Remove unnecessary parentheses.

$2 (x + 9) (x - 9) (x^{2} - 2) = 0$