Question 1072807: Solve the equation 6z2−7z−3=0
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Write your answers as a list of integers or reduced fractions, with your answers separated by (a) comma(s).
Found 2 solutions by Fombitz, Theo:
Answer by Fombitz(32388)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! your equation is:
6z^2 - 7z - 3 = 0
since this is in standard form, you get:
a = 6
b = -7
c = -3
a is the coefficient of the z^2 term.
b is the coefficient of the z term.
c is the constant term.
you would factor this using the quadratic formula, but that's not necessary in this case because the equation can be factored manually.
there is a method that splits the middle term into constituent parts which we'll use.
start with:
6z^2 - 9z - 3 = 0
multiply 6 * 3 to get 18.
since the sign of the z term and the sign of the constant term are both negative, then you will need to subtract the factors of 18 to get the constituent middle terms.
the process is as follows:
1 * 18 is one of the factors, but if you subtract 1 from 18 or 18 from 1, you will not get -7
your next factors are 2 * 9.
if you subtract 2 from 9, you will get 7 which is not equal to -7.
if you subtract 9 from 2, you will get -7 which is equal to -7.
your middle term will be separated into -9z and 2z
your equation of 6z^2 - 9z - 3 = 0 becomes:
6z^2 - 9z + 2z - 3 = 0
group the left two terms and the right two terms together to get:
(6z^2 - 9z) + (2z - 3) = 0
you now want to factor these 2 terms so that there is a common factor between them.
you will get:
3z * (2z - 3) + (2z - 3) = 0
this is the same as:
3z * (2z - 3) + 1 * (2z - 3) = 0
the common term between them is (2z - 3).
factor this out to get:
(3z + 1) * (2z - 3) = 0
those are your factors.
if you multiply those factors out, you will get 6z^2 - 7z - 3 = 0
if you had used the quadratic formula, you would have gotten the same answer.
now that you have the factors, you can solve for z.
start with:
(3z + 1) * (2z - 3) = 0
this equation will be true if (3z + 1) = 0 or (2z - 3) = 0 or if both of them are equal to 0.
therefore, to find your solution, just set each factor equal to 0 and solve.
start with 3z + 1 = 0
subtract 1 from both sides to get 3z = -1
divide both sides by 3 to get z = -1/3
start with 2z - 3 = 0
add 3 to both sides to get 2z = 3
divide both sides by 2 to get z = 3/2
your solutions are:
z = -1/3 or z = 3/2
there are various methods to factor, some of which are covered in the following tutorials.
http://www.purplemath.com/modules/factquad.htm
http://www.purplemath.com/modules/quadform.htm
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