Question

How do you solve `6x^2+7x+2=0`?

Answer 1

`x = -1/2` and `x = -2/3`

Explanation:

You need to factor this quadratic by playing with multipliers for 6 (1x6, 2x3, 3x2, 6x1) and multipliers for 2 (1x2, 2x1):

`(2x + 1)(3x + 2) = 0`

Now we can solve each term for `0`:

`2x + 1 = 0`

`2x + 1 - 1 = 0 - 1`

`2x = -1`

`(2x)/2 = -1/2`

`x = -1/2`

and

`3x + 2 = 0`

`3x + 2 - 2 = 0 - 2`

`3x = -2`

`(3x)/3 = -2/3`

`x = -2/3`

Answer 2

`x = -1/2" "` or `" "x = -2/3`

Explanation:

Given:

`6x^2+7x+2 = 0`

Use an AC method:

Find a pair of factors of `AC = 6*2 = 12` with sum `B = 7`.

The pair `4, 3` works.

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

`0 = 6x^2+7x+2`

`color(white)(0) = 6x^2+4x+3x+2`

`color(white)(0) = (6x^2+4x)+(3x+2)`

`color(white)(0) = 2x(3x+2)+1(3x+2)`

`color(white)(0) = (2x+1)(3x+2)`

Hence zeros: `x = -1/2` and `x = -2/3`