factorize completely :
12x squared-23x+5
This kind of thing just takes practice. You know the factors will look like one of
(12x-?)(x-?)
(6x-?)(2x-?)
(4x-?)(3x-?)
where the two ? numbers are 1 and 5. Then it's just trial and error. Eventually you will wind up with
12x^2-23x+5
(4x-1)(3x-5)
thanks much, God bless you
To factorize the quadratic expression 12x² - 23x + 5 completely, we need to find two binomial factors that, when multiplied, result in the original expression.
Step 1: Multiply the coefficient of the quadratic term (12) by the constant term (5). The result is 60.
Step 2: Look for two numbers whose product is 60 and whose sum is equal to the coefficient of the linear term (-23). The numbers that satisfy these conditions are -20 and -3.
Step 3: Rewrite the linear term (-23x) using the found numbers. Replace -23x with -20x - 3x.
12x² - 20x - 3x + 5
Step 4: Group the terms two at a time and factor by grouping.
(12x² - 20x) + (-3x + 5)
Step 5: Factor out the greatest common factor from each group.
4x(3x - 5) - 1(3x - 5)
Step 6: Observe that we now have a common binomial factor, (3x - 5), in both groups.
(4x - 1)(3x - 5)
Therefore, the expression 12x² - 23x + 5 is completely factorized as (4x - 1)(3x - 5).