How do you factor examples b), e) and f) below?
b) #y = 8x^2-10x-3#
e) #y = 5x^3-80x#
f) #y = 36x^2-1#
b)
e)
f)
1 Answer
b)
e)
f)
Explanation:
Example b)
#y = 8x^2-10x-3#
This quadratic is in the form
In order to tell whether it factors 'nicely', let us first check the discriminant:
#Delta = b^2-4ac = (-10)^2-4(8)(-3) = 100+96 = 196 = 14^2#
Since this is a perfect square, the given quadratic will factor exactly.
Let's use an AC method to find the factors:
Look for a pair of factors of
The pair
Use this pair to split the middle term and factor by grouping:
#y = 8x^2-10x-3#
#color(white)(y) = 8x^2-12x+2x-3#
#color(white)(y) = (8x^2-12x)+(2x-3)#
#color(white)(y) = 4x(2x-3)+1(2x-3)#
#color(white)(y) = (4x+1)(2x-3)#
Example e)
#y = 5x^3-80x#
For this example, we can separate out the common factor
#a^2-b^2 = (a-b)(a+b)#
with
#y = 5x^3-80x#
#color(white)(y) = 5x(x^2-16)#
#color(white)(y) = 5x(x^2-4^2)#
#color(white)(y) = 5x(x-4)(x+4)#
Example f)
#y = 36x^2-1#
Notice that both the terms are perfect squares, so we can use the difference of squares identity again...
#y = 36x^2-1 = (6x)^2-1^2 = (6x-1)(6x+1)#