Given that the absolute value of the difference of the two roots of ax^2 + 5x - 3 = 0 is \(7/2\), and a is positive, what is the value of a?
+128448 Let the roots be m and n
The sum of the roots = -5/a = m + n
Square both sides of this
25/a^2 = m^2 + 2mn + n^2
The product of the roots = - 3/a = mn
Then -4(-3)/a = -4mn
So 12/a = -4mn
So
25/a^2 + 12/a = m^2 + 2mn - 4mn + n^2
25/a^2 + 12/a = m^2 - 2mn + n^2
25/a^2 + 12/a = (m +n) (m - n)
25/a^2 + 12/a = (-5/a)(m - n) multiply through by a
25/a + 12 = -5 ( m - n)
If we let m - n = -7/2 then we have that
25/a + 12 = -5(-7/2)
25/a + 12 = 35/2 multiply through by 2
50/a + 24 = 35
50 / a = 21
a = 50 / 21
