Basic Math Examples

${(a^{2} - 2 a)}^{2} - 23 (a^{2} - 2 a) + 120$

Step 1

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

$u^{2} - 23 u + 120$

Step 2

Factor $u^{2} - 23 u + 120$ using the AC method.

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Step 2.1

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 $120$ and whose sum is $- 23$ .

$- 15, - 8$

Step 2.2

Write the factored form using these integers.

$(u - 15) (u - 8)$

Step 3

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

$(a^{2} - 2 a - 15) (a^{2} - 2 a - 8)$

Step 4

Factor $a^{2} - 2 a - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 4.2

Write the factored form using these integers.

$(a - 5) (a + 3) (a^{2} - 2 a - 8)$

Step 5

Factor.

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Step 5.1

Factor $a^{2} - 2 a - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 5.1.2

Write the factored form using these integers.

$(a - 5) (a + 3) ((a - 4) (a + 2))$

Step 5.2

Remove unnecessary parentheses.

$(a - 5) (a + 3) (a - 4) (a + 2)$