Question 157288
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}}}

<pre><font size = 4 color = "indigo"><b>
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:

{{{am^2 + bm + c = 0}}}, {{{ax^2+bx+c=0}}} or whatever 
the letter of the unknown happens to be:

That's the only way you can tell what to substitute for
{{{a}}}, {{{b}}}, and {{{c}}} in the formula:

{{{DISCRIMINANT=b^2-4ac}}}

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For your problem 35:

{{{4m^2+25 = 20m}}}

You must first get a 0 on the right side by adding {{{-20m}}} to
both sides:

{{{4m^2+25-20m=20m-20m}}}

{{{4m^2+25-20m=0}}}

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

{{{4m^2-20m+25=0}}}

Then when you compare that to

{{{am^2 + bm + c = 0}}}, 

it is easy to see that {{{a=4}}},{{{b=-20}}}, and {{{c=25}}}.

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

{{{DISCRIMINANT=b^2-4ac}}}

{{{DISCRIMINANT=(-20)^2-4(4)(25)}}}

{{{DISCRIMINANT=400-400}}}

{{{DISCRIMINANT=0}}}

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:

{{{x^2=x}}}

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

{{{x^2-x=x-x}}}

{{{x^2-x=0}}}

But the left side contains only two terms! So you must 
add on a {{{0}}} to the left side:

{{{x^2-x+0=0}}}

Also it may be helpful to write the understood {{{1}}}'s before
{{{x^2}}} and {{{x}}}, like this:

{{{1x^2-1x+0=0}}}

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

{{{ax^2 + bx + c = 0}}}, 

it is easy to see that {{{a=1}}},{{{b=-1}}}, and {{{c=0}}}.

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

{{{DISCRIMINANT=b^2-4ac}}}

{{{DISCRIMINANT=(-1)^2-4(1)(0)}}}

{{{DISCRIMINANT=1-0}}}

{{{DISCRIMINANT=1}}}

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

Edwin</pre>