Hi, thanks a lot for this website which is proving extremely helpful! On the last line, why is it that (m-3) gives m >3 but (m-4)<0 gives m<4?
(m-3)< 0 gives m>3
i dont know why that happened twice, sorry
We know that the quadratic (m-3)(m-4) = 0 when m = 3 or m = 4. But we are interested in the values of m for which it is less than 0. If you sketch the graph you’ll see that, when m is large and negative the quadratic is large and positive, when m is large and positive the quadratic is large and positive, and the quadratic is negative when 3 < m < 4.
I am sorry, but why were you able to just cross off the 16 from in front of the parentheses in line 3? The same applies to subtracting the 4 from the first parentheses. Doesn´t that change the results of m?
Line 3 is transformed into line 4 by dividing both sides of the inequality by 16. That’s why the 16 disappears, and why the (8-4m)^2 becomes (2-m)^2.
Hi, I don’t understand how you got from the equation on the first line to the inequality on the second? where did all the x values go and how did the first bracket become squared?
The first line has a quadratic in the form ax^2 + bx +c = 0.
The second line is equivalent to b^2 – 4ac < 0, which is the condition for the quadratic in the first line to have no real roots.
How did you get that top line to start with?
I just took the equation in the question, subtracted m(4x-3) from both sides, and simplified to create the ax^2 + bx + c = 0 form.
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