Precalculus Examples

${(x + 2)}^{2} - (x + 2) - 42$

Step 1

Let $u = x + 2$ . Substitute $u$ for all occurrences of $x + 2$ .

$u^{2} - u - 42$

Step 2

Factor $u^{2} - u - 42$ 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 $- 42$ and whose sum is $- 1$ .

$- 7, 6$

Step 2.2

Write the factored form using these integers.

$(u - 7) (u + 6)$

Step 3

Replace all occurrences of $u$ with $x + 2$ .

$(x + 2 - 7) (x + 2 + 6)$

Step 4

Simplify.

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Step 4.1

Subtract $7$ from $2$ .

$(x - 5) (x + 2 + 6)$

Step 4.2

Add $2$ and $6$ .

$(x - 5) (x + 8)$