Question

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

Answer 1

`6x^2-23x+15=(6x-5)(x-3)`

Explanation:

Use an AC method:

Find a pair of factors of `AC=6*15=90` with sum `B=23`

The pair `18, 5` works in that `18xx5 = 90` and `18+5=23`.

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

`6x^2-23x+15`

`=6x^2-18x-5x+15`

`=(6x^2-18x)-(5x-15)`

`=6x(x-3)-5(x-3)`

`=(6x-5)(x-3)`

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

`a^2-b^2=(a-b)(a+b)`

with `a=(12x-23)` and `b=13` as follows:

I will multiply by `4*6 = 24` first to avoid some fractions:

`24(6x^2-23x+15)`

`=144x^2-552x+360`

`=(12x)^2-2(12x)(23)+360`

`=(12x-23)^2-23^2+360`

`=(12x-23)^2-529+360`

`=(12x-23)^2-169`

`=(12x-23)^2-13^2`

`=((12x-23)-13)((12x-23)+13)`

`=(12x-36)(12x-10)`

`=(12(x-3))(2(6x-5))`

`=24(x-3)(6x-5)`

Dividing both ends by `24` we find:

`6x^2-23x+15 = (x-3)(6x-5)`