Question

How do I solve this quadratic equation?

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

Answer 1

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

Explanation:

`6x^2 + 7x + 2`

can be factored into a binomial,

`(3x+3/2)(2x+4/3)`

By setting a factor to zero we can solve for an x value
`3x+3/2 = 0`
`x = -1/2`

`2x+4/3 = 0`
`x= -2/3`

Answer 2

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

Explanation:

We can solve this quadratic with the strategy factoring by grouping. Here, we will rewrite the `x` term as the sum of two terms, so we can split them up and factor. Here's what I mean:

`6x^2+color(blue)(7x)+2=0`

This is equivalent to the following:

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

Notice, I only rewrote `7x` as the sum of `3x` and `4x` so we can factor. You'll see why this is useful:

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

We can factor a `3x` out of the red expression, and a `2` out of the orange expression. We get:

`color(red)(3x(2x+1))+color(orange)(2(2x+1))=0`

Since `3x` and `2` are being multiplied by the same term (`2x+1`), we can rewrite this equation as:

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

We now set both factors equal to zero to get:

`3x+2=0`

`=>3x=-2`

`color(blue)(=>x=-2/3)`

`2x+1=0`

`=>2x=-1`

`color(blue)(=>x=-1/2)`

Our factors are in blue. Hope this helps!

Answer 3

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

Explanation:

Hmm...
We have:
`6x^2+7x+2=0` Since `x^2` is being multiplied by a number here, let's multiply `a` and `c` in `ax^2+bx+c=0`

`a*c=6*2=>12`

We ask ourselves: Do any of the factors of `12` add up to `7`?

Let's see...

`1*12` Nope.

`2*6` Nope.

`3*4` Yep.

We now rewrite the equation like the following:

`6x^2+3x+4x+2=0` (The order of `3x` and `4x` does not matter.)

Let's separate the terms like this:

`(6x^2+3x)+(4x+2)=0` Factor each parenthesis.

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

For better understanding, we let `n=2x+1`

Replace `2x+1` with `n`.

`=>3xn+2n=0` Now, we see that each group have `n` in common.

Let's factor each term.

`=>n(3x+2)=0` Replace `n` with `2x+1`

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

Either `2x+1=0` or `3x+2=0`

Let's solve each case.

`2x+1=0`

`2x=-1`

`x=-1/2` That's one answer.

`3x+2=0`

`3x=-2`

`x=-2/3` That's another.

Those two are our answers!