Answer:
Step-by-step explanation:
1) GCF = Greatest common factor which would be 6x so lets get started.
[tex]6x^3+36x+48=0\\6x(x^2+6x+8)=0\\x^2+6x+8=0\\[/tex]
now we solve this with by quadratic formula.
[tex]x= -\frac{b+-\sqrt{b^2-4ac} }{2a}[/tex]
so a=1 , b=6, c =8
and we get two values of x which is
[tex]x=-2 \\and \\x=-4[/tex]
but in the question it says we will end up with 3 solutions. Our third solution is actually [tex]x=0[/tex]
because we took 6x common as the greatest common factor so
[tex]6x=0\\x=0[/tex]
so now we have 3 solutions
[tex]x=-4\\x=-2\\x=0[/tex]
2) (x-9)(4x+1)=0
since this equation is already in root form we equate each root to 0
[tex](x-9)(4x+1)=0\\x-9=0\\x=9 \\4x+1=0\\4x=-1\\x=-1/4[/tex]
so our solutions are x=9 and x=-1/4
3)
[tex](x+5)^2-12=132\\(x+5)^2=132+12\\(x+5)^2=144\\[/tex]
now since we need to solve using square roots we apply square roots on both sides
[tex]\sqrt{(x+5)^2}=\sqrt{144}\\x+5=+12 \\x+5=-12[/tex]
since when we take the square root of a number we get positive and negative of that number example here
x+5=±12
so
x+5=12 and x+5=-12
x=7 and x=-17
4) Difference of two squares means we just need to factorize it in root form so lets begin,
[tex]4x^2-49[/tex]
now if we see this expression can be written like this as well
[tex](2x)^2-(7)^2[/tex]
and if we expand this into the difference formula which is
[tex](a-b)(a+b)\\[/tex]
we get
[tex]4x^2-49=(2x-7)(2x+7)\\[/tex]
where a is 2x and b is 7
