Question

How do you factor the expression `2x^2 - 5x - 3`?

Answer 1

`2x^2-5x-3 = (2x+1)(x-3)`

Explanation:

Use an AC method. Look for a pair of factors of `AC = 2*3 = 6` with difference `5`.

The pair `6, 1` works, in that `6xx1=6` and `6-1=5`

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

`2x^2-5x-3`

`= 2x^2-6x+x-3`

`= (2x^2-6x)+(x-3)`

`= 2x(x-3)+1(x-3)`

`= (2x+1)(x-3)`

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Alternative method

We will use the difference of squares identity which can be written:

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

with `a=(4x-5)` and `b=7` later.

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We will follow the following steps:

1. Multiply by `2^3 = 8` to simplify the later arithmetic.
2. Complete the square to make a difference of squares.
3. Factor the difference of squares.
4. Simplify.
5. Divide by `8`.

First multiply by `8`:

`8 xx (2x^2-5x-3) = 16x^2-40x-24`

Then:

`16x^2-40x-24`

`=(4x-5)^2-5^2-24`

`=(4x-5)^2-49`

`=(4x-5)^2 - 7^2`

`=((4x-5)-7)((4x-5)+7)`

`=(4x-12)(4x+2)`

`=(4(x-3))(2(2x+1))`

`=8(x-3)(2x+1)`

Then divide by `8` to find:

`2x^2-5x-3 = (x-3)(2x+1)`