quadratic equations (FORM4)

1.
If α and β are the roots of quadratic equation 5х² - 7х + c = 0, and α : β = 3 :4. find the possible value (s) of c.

Let 3k and 4k be α and β respectively.

Sum of the roots:
α + β = 7/5
3k + 4k = 7/5
k = 1/5
Then, α = 3/5 and β = 4/5

Product of the roots:
αβ = c/5
(3/5) * (4/5) = c/5
c = 12/5


= = = = =
2.
If the difference between the roots of 4х² + 2kx - 5 = 0 is 3, find thepossible value (s) of k.

Let α be the smaller root.
Then, the greater root = α + 3

Product of the roots:
α(α + 3) = -5/4
α² + 3α = -5/4
4α² + 12α + 5 = 0
(2α + 1)(2α + 5) = 0
α = -1/2 or α = -5/2

When α = -1/2, α + 3 = 5/2
Sum of the roots:
(-1/2) + (5/2) = -2k/4
k = -4

When α = -5/2, α + 3 = 1/2
Sum of the roots:
(-5/2) + (1/2) = -2k/4
k = 4

Hence, k = 4 or k = -4