How do I solve this quadratic equation?

6x2+7x+2=0

3 Answers
Mar 9, 2018

x=12 and x=23

Explanation:

6x2+7x+2

can be factored into a binomial,

(3x+32)(2x+43)

By setting a factor to zero we can solve for an x value
3x+32=0
x=12

2x+43=0
x=23

Mar 9, 2018

x=12,23

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:

6x2+7x+2=0

This is equivalent to the following:

6x2+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:

6x2+3x+4x+2=0

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

3x(2x+1)+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

x=23

2x+1=0

2x=1

x=12

Our factors are in blue. Hope this helps!

Mar 9, 2018

12=x=23

Explanation:

Hmm...
We have:
6x2+7x+2=0 Since x2 is being multiplied by a number here, let's multiply a and c in ax2+bx+c=0

ac=6212

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

Let's see...

112 Nope.

26 Nope.

34 Yep.

We now rewrite the equation like the following:

6x2+3x+4x+2=0 (The order of 3x and 4x does not matter.)

Let's separate the terms like this:

(6x2+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=12 That's one answer.

3x+2=0

3x=2

x=23 That's another.

Those two are our answers!