Melody

The middle one should be easy.  I'm going to give hints.

How do you find the roots?

   you have to simultaneously solve y=0 with that parabola. That will give the points where the p[arabola cuts the x axis.

The vertex is on the axis of symmetry and the axis of symm is half way between the roots.

Once you have the x value you can find the y with substitution.

the roots are 

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)      seperate those 2 out and you will see that they are symmetrical about a very easy to find point. 

Usually about a week I think.

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Hi, I like it when people ask for more explanation.   

The very first line is   M=4D

If I divide both sides by 2 then I get.

\(\frac{1}{2}M=2D\)

Sounds right to me :)

Don't be so lazy. TYPE just one of these questions in, lean from the answer, and do the others yourself.

You won't learn anything from copying all of other peoples answers. 

Two line segments, AB and CD, bisect one another at point 0. Prove that the lines AC and BD are parallel.

I suggest you start with the diagram.

Consider triangles OAC and  OBD

OA=OB  Since 0 bisects AB

OC=OD  since o bisects CD

angleBOD=angleAOC     Vertically opposite angles

therefore

triangle OAC is congruent to triangle OBD        Two sides and included angle test.

therefore

angle OAC = angle OBC      corresponding angles in congruent triangles

Now consider AB as the transversal that cuts the lines AC and BD

angle OAC  angle OBC are alternate angles and since they are equal  AC must be parallel to BD

If a line meets two parallel lines and is perpendicular to one of them, prove that it is perpendicular to the other.

If two parallel lines are cut by a transversal then the cointerior angles are supplementary.

(If one is x the other is 180-x)

If one is 90 then thte other is 180-90=90.

So if the transversal is perpendicular to one of the lines it must be perpendicular to the other as well.

I have never seen notation like this.

 f[-4(x=1)]  looks super odd.

Maybe another mathematician has seen something like this before?

By the way, your heading does not inspire me to answer at all.   I am just interested in the question.

I like the 'please' but the word 'fast' inspires me to answer more slowly.

I am just telling you this because you said 'please' and I am giving you the opportunity to ask differently next time.

Is there a picture to go with this?  It doesn't make much sense.