Question

Number ranges + quadratic equations

Answer 1

Find the range (or ranges) of possible values of the real number a for all real values of x.

\[a^2x^2 +2(2a - 5)x +8 > 0\]

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This is the second part of a question, (x-2 is one of the roots) for the first part I figured out that this equation was made up of the following factors:

\[(9x-4)(x-2)\]

or

\[(x-4)(x-2)\]

As well as the fact that "a" can equal 1, -3

Answer 2

Therefore, I calculated that (would like it if you could check this for me please!)

in order for the equation to be greater than 0,

x<2 or x>4

x<4/9 or x>2

Unfortunately, I'm stuck there. I'm not sure how to find the ranges of "a" for which this is applicable.

Thanks in advance! c:

Answer 3

well wouldn't you use the dircriminant to find where

\[b^2 - 4ac \ge 0\]

as the solutions need to real... but they can be irrational

Answer 4

Ah yes. Good idea! I had this idea that everything needed to be super complicated. =/

I'm going to close this question, I missed such a basic piece of information it's embarrassing haha.

Thanks again! c: