Algebra - Finding the deffierence between roots of an equation?

Question

Hi, as part or a set of algebra questions from school, I have been asked

"Find the values of m for which the difference between the roots of the equation 16x^2 = mx - 2501 is 5."

I know how to use the discriminant to find the values that are real roots etc., but I have no idea how to work out the values in which the roots have a difference of 5.

The answers start with

"Let y, z be the 2 roots.
If ax^2 +bx +c = 0 then
y + z = -b / a
yz = c / a "
I have no idea what that means?
ANy hints would be appreaciated

Answer 1

METHOD 1:

Using their method:

Let y and z be the 2 roots.

That means a(x - y)(x - z) = 0 will be the equation.
Expanding out you would have:
a(x² - zx - yx + yz) = 0

ax² + a(-z - y)x + ayz = 0

Now it should be easy to map:

a = a

b = a(-z - y)
b = -a(y + z)
y + z = b / -a

c = ayz
yz = c / a

METHOD 2:

Using the quadratic formula:

16x² - mx + 2501 = 0

Quadratic formula tells us the roots are:

-b +/- sqrt( b² - 4ac )
----------------------------
........... 2a

If you subtract these two roots you'll get:
2( sqrt (b² - 4ac) )
-------------------------
2a

sqrt (b² - 4ac)
-------------------
a

Plugging in a = 16, b = -m, c = -2501

sqrt( (-m)² - 4(16)(2501) )
-----------------------------------
16

Solve for m from here

Answer 2

I assume you know the quadratic formula for the two roots of a quadratic equation,
ax^2+bx+c=0
then
x=(-b+sqrt(b^2-4ac))/2a or (-b-sqrt(b^2-4ac))/2a
The sum of the roots will then be the sum of the two numbers above. Add them and you get
-b/a
Multiplying them is somewhat harder, but you will end up with
c/a
Now for your question, a=16, b=-m, and c=2501 are the coefficients. Let y and z be the two roots, then
y+z = -b/a = m/16
and
yz = c/a = 2501/16
What value for m would make y-z=5 or z-y=5?(they are the same)

Answer 3

Try aligining the question
16x^2=mx-2501
16x^2-mx+2501=0
16=a -m=b 2501=c
Now.
Y + z = -b / a
Yz = c / a

y= (-b/a) - z
y= (m/16) - z
Therefore ((m/16)-z)z= c/a
((m/16)-z)z= 2501/16
(m/16)-z= 2501/16z
m/16 = (2501/16z) +z
m/16 = (2501+16z^2)/16z
m=(2501+16z^2)/z
im not sure if thats right im just thinking as im going, hopefully its triggered somthing