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!