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How do you factor $7 k^{2} + 19 k + 10$ $7k^2+19k+10$ ?

1 Answer

Apr 5, 2016

Look for factor pairs of $7$ $7$ and of $10$ $10$ which when cross multiplied equal $19$ $19$ .
This should allow you to find:
$XXX (k + 2) (7 k + 5)$ $color(white)("XXX")(k+2)(7k+5)$

Explanation:

Note that since all terms in the given expression are positive, we only need to consider positive factors.

Factors of $7$ $7$
$XXX {(1, 7)}$ $color(white)("XXX"){color(red)(color(white)("")(1,7))}$
Factors of $10$ $10$
$XXX {(1, 10), (2, 5)}$ $color(white)("XXX"){color(blue)(color(white)("")(1,10),(2,5))}$

Using $X$ $" X "$ to indicate cross multiplication:
Since $1 \times 5 + 7 \times 2 = 19$ $color(red)(1)xxcolor(blue)(5)+color(red)(7)xxcolor(blue)(2) = 19$

$color(white)("XXX"){:(color(red)(1)),(color(red)(7)):}" X "{:(color(blue)(2)),(color(blue)(5)):}=19$

The required factors are
$color(white)("XXX")(color(red)(1)k+color(blue)(2))(color(red)(7)k+color(blue)(5))$

Note that this method only works since the coefficients are integer.

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