Factorize $x^{2} + 5 x + 6$

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Solution

Comparing $x^{2} + 5 x + 6$ with $x^{2} + (a + b) x + a b$
We have, $a b = 6, a + b = 5$ and $x = x$ .
If $a b = 6$ , it means $a$ and $b$ are factors of $6$ .
Let us try with $a = 2$ and $b = 3$ . These values satisfy $a b = 6$ and $a + b = 5$ .
Therefore the pair of values $a = 2$ $a n d$ b = 3$ is the right choice.
Using, $x^{2} + (a + b) x + a b = (x + a) (x + b)$
$x^{2} + 5 x + 6 = x^{2} + (2 + 3) x + (2 \times 3)$
$= (x + 2) (x + 3)$
$∴ (x + 2)$ and $(x + 3)$ are the factors of $x^{2} + 5 x + 6$ .

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According to factor theorem, x-6 is the factor of $x^{2} - 5 x - 6$

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