# How do you factor the expression  6x^2 - 23x + 15?

May 8, 2016

$6 {x}^{2} - 23 x + 15 = \left(6 x - 5\right) \left(x - 3\right)$

#### Explanation:

Use an AC method:

Find a pair of factors of $A C = 6 \cdot 15 = 90$ with sum $B = 23$

The pair $18 , 5$ works in that $18 \times 5 = 90$ and $18 + 5 = 23$.

Use this pair to split the middle term and factor by grouping:

$6 {x}^{2} - 23 x + 15$

$= 6 {x}^{2} - 18 x - 5 x + 15$

$= \left(6 {x}^{2} - 18 x\right) - \left(5 x - 15\right)$

$= 6 x \left(x - 3\right) - 5 \left(x - 3\right)$

$= \left(6 x - 5\right) \left(x - 3\right)$

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Alternatively, you can complete the square and use the difference of squares identity:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

with $a = \left(12 x - 23\right)$ and $b = 13$ as follows:

I will multiply by $4 \cdot 6 = 24$ first to avoid some fractions:

$24 \left(6 {x}^{2} - 23 x + 15\right)$

$= 144 {x}^{2} - 552 x + 360$

$= {\left(12 x\right)}^{2} - 2 \left(12 x\right) \left(23\right) + 360$

$= {\left(12 x - 23\right)}^{2} - {23}^{2} + 360$

$= {\left(12 x - 23\right)}^{2} - 529 + 360$

$= {\left(12 x - 23\right)}^{2} - 169$

$= {\left(12 x - 23\right)}^{2} - {13}^{2}$

$= \left(\left(12 x - 23\right) - 13\right) \left(\left(12 x - 23\right) + 13\right)$

$= \left(12 x - 36\right) \left(12 x - 10\right)$

$= \left(12 \left(x - 3\right)\right) \left(2 \left(6 x - 5\right)\right)$

$= 24 \left(x - 3\right) \left(6 x - 5\right)$

Dividing both ends by $24$ we find:

$6 {x}^{2} - 23 x + 15 = \left(x - 3\right) \left(6 x - 5\right)$