How do I solve this quadratic equation?

6x^2 + 7x +2=0

3 Answers
Mar 9, 2018

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

Mar 9, 2018

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!

Mar 9, 2018

-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!