Assuming 6x 2 - 7x - 3 = 0, find the possible values of x using factorization method.
Solution.
6x 2 - 7x - 3 = 0
Step1
As we can see in the equation 6x 2 - 7x - 3 = 0, the coefficient of x 2 is 6 and not 1.
The first step will therefore be to multiply the coefficient of x 2 (ie 6) and the constant term (ie -3)
6 x -3 = -18
Step2
Find two numbers which, when multiplied will give -18 and when added (or subtracted) will give the coefficient of x (ie -7)
Factors: factors of -18 are
18 and -1 (18 x -1 = -18; 18 - 1 = +17. This does not satisfy the equation)
9 and -2 (9 x -2 = -18; 9 - 2 = +7. This does not satisfy the equation)
-9 and +2 (-9 x 2 = -18; -9 + 2 = -7. This satisfies the equation)
Step3
In the equation 6x 2 - 7x - 3 = 0, replace -7x with -9x and +2x in the equation.
We then have 6x 2 - 9x + 2x - 3 = 0 (by splitting -7x into -9x and 2x)
Step4
Group the first two terms together and the last two terms together as follows
6x 2 - 9x + 2x - 3 = 0
By bringing out common terms, this becomes
3x(2x - 3) + 1(2x - 3) = 0
Since the two brackets are the same, we take one of them and multiply by the coefficients of the brackets as follows
(2x - 3)(3x + 1) = 0
If two things are multiplied and the result is zero, it means that either one of them is equal to zero or both are equal to zero
Therefore,
Either 2x - 3 = 0, which gives 2x = 0 + 3
2x = 3 and therefore x = 3/2
Or 3x + 1 = 0, which gives 3x = 0 - 1
3x = -1 and therefore x = -1/3
The values of x that satisfy the equation
6x 2 - 7x - 3 = 0 are x = 3/2 or -1/3
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