If α and β are the roots of the equation 4x^2 - 5x + 2 = 0 . Find the equation whose roots are ( α + 3β ) and ( 3α - β ).
Answers
4x^2 - 5x + 2 = 0
α + β = 5/4
α β = 2/4 = 1/2
α = 5/4 - β
β = 5/4 - α
α β = 1/2
α = 1/(2β)
β = 1/(2α)
α + 3β + 3α + β
= 4α + 4β
= 4(α + β)
= 4 * 5/4
= 5
(α + 3β)(3α + β)
= (α + 15/4 - 3α)(15/4 - 3β + β)
= (15/4 - 2α)(15/4 - 2β)
= (15 - 8α)/4 * (15 - 8β)/4
= (225 - 120α - 120β + 64αβ)/16
= [225 - 120(α + β) + 64αβ]/16
= (225 - 120 * 5/4 + 64 * 1/2)/16
= (225 - 150 + 32)/16
= 107/16
= (5/4 - β + 3β) (3α + 5/4 - α)
= (2β + 5/4) (2α + 5/4)
= (8β + 5)/4 * (8α + 5)/4
= (64αβ + 40α + 40β + 25)/16
= [64αβ + 40(α + β) + 25]/16
= (64 * 1/2 + 40 * 5/4 + 25)/16
= (32 + 50 + 25)/16
= 107/16
Why can't they make one up that gives a whole number??
Anyhow we have been using the quadratic's sum and products of roots concept
as follow:
ax^2 + bx + c = 0
A quadratic equation can also be represented in the form :
ax^2 + (sum of roots)x + (product of roots) = 0
Formula for the sum of roots: －b/a
Formula for the product of roots: c/a
So in this case the quadratic equation will be:
x^2 - 5x + 107/16 = 0
or:
16x^2 - 80x + 107 = 0
To check this I used the actual roots of the both equations to see if this meets the required condition:
α = 1/8 (5 - i √ 7) , β = 1/8 (5 + i √ 7)
And for the 2nd equation we just found:
α + 3β = (5/2 - i √ 7/4), 3α + β = (5/2 + i √ 7/4)
Now let's check:
α + 3β = 1/8 (5 - i √ 7) + 3/8 (5 + i √ 7) = (5/2 + i √ 7/4)
3α + β = 3/8 (5 - i √ 7) + 1/8 (5 + i √ 7) = (5/2 - i √ 7/4)
Answer:
Hope this helps you
Mark as Branliest and follow me
