SOLUTION: How do I find the discriminant? The question ask to find {{{b^2-4ac}}} and the number of real solutions to each equation. 35. {{{4m^2+25 = 20m}}} 45. {{{x^2=x}}}

You must first rearrange the equations so that 0 appears
on the right and the three terms on the left are in 
descending order like this:

,  or whatever 
the letter of the unknown happens to be:

That's the only way you can tell what to substitute for
, , and  in the formula:



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For your problem 35:



You must first get a 0 on the right side by adding  to
both sides:





And you must write the left side in descending order, like 
this:



Then when you compare that to

, 

it is easy to see that ,, and .

So then you can easily substitute those values in the 
expression for the discriminant:









When the discriminant is 0, there is 
exactly 1 real solution. So this particular
quadratic equation has 1 real solution.


If it were negative there would be no real solutions,
and if it were positive there would be exactly two real
solutions.

---------------------------------

For your problem 45:



You must first get a 0 on the right side by adding  to
both sides:





But the left side contains only two terms! So you must 
add on a  to the left side:



Also it may be helpful to write the understood 's before
 and , like this:



The left side is already in descending order, so when you 
compare that to

, 

it is easy to see that ,, and .

So then you can easily substitute those values in the 
expression for the discriminant:









Since the discriminant is a positive number,
there are exactly two real solutions.

Edwin