Question

If a and b are the roots of the equation 4x²-5x+2=0,find the equation whose roots are a+3b and 3a+b.

Answer 1

a & b are roots of the equation ;

4x^2 - 5x + 2 = 0

Sum of Roots
= a+b = 5/4

Product of Roots
= ab = 1/2
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In the quadratic equation whose roots are ;
a+3b & 3a+b

Sum of Roots
= a+3b+3a+b
= 4a+4b
= 4(a+b)
= 4×(5/4)
= 5
_____________________
Product of Roots
= (a+3b) (3a+b)
= 3a^2 + ab + 3ab + 3b^2
= 3(a^2+b^2) + 4ab
= 3× (a+b)^2 - 2ab + 4ab

= 3 × (a+b)^2 + 2ab
= 3 × (5/4)^2 + 2×(1/2)
= 75/16 + 1
= 91/16
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The required quadratic equation is ;

x^2 - (sum of roots) x + (product of roots) = 0

x^2 - 5x + 91/16 = 0

16x^2 - 80x + 91 = 0
_____________________

Answer 2

4x^2 -5x +2 =0

sum of roots = a + b = 5/4


products of roots = ab = 2/4 = 1/2

now,

equation of given, roots (a +3b ) and (3a + b)

x^2 -(sum of roots )x + products of roots =0

sum of roots = (a +3b) +(3a + b) = 4(a +b ) = 5

products of roots = ( a+3b) (3a + b)
= 3a^2 + ab +9ab + 3b^2
=3{(a + b)^2 -2ab} +10ab
= 3(a + b)^2 +4ab
=3(25/16) +4 (1/2)
=75/16 +2. = (75+32)/16 = 107/16


now,

x^2 -(5)x + 107/16 =0