Question

Write the quadratic equation that has roots (√3+1)/2 and (√3 -1 )/2, if its coefficient with x^2 is equal to:

a) 1
b) 5
c) - 1/2
d) √3

Answer 1

The equation is:
`x^2-x+(1)/(2)`

(A)
`x^2-x+(1)/(2)=0`

(B)
`5x^2-5x+(5)/(2)=0`

(C)
`-(1)/(2)`

`-(1)/(2)x^2+(1)/(2)x-(1)/(4)=0`

d) √3

`(√(3)) x^2-x√(3)+(√(3))/(2)=0`

Explanation:

If the roots of the quadratic equation are:
`(√(3)+1)/(2) and (√(3)-1)/(2)`

This means:

x=
`(√(3)+1)/(2)` OR
`x= (√(3)-1)/(2)`

x-
`x-(√(3)+1)/(2)=0 \: or\: x- (√(3)-1)/(2)=0`=0 or x-

If a=0 or b=0, then ab=0

Therefore:

`(x-(√(3)+1)/(2))(x-(√(3)-1)/(2))=0\\x^2-(x(√(3)-1))/(2)-(x(√(3)+1))/(2)+((√(3)-1))/(2))((√(3)+1)/(2))=0\\x^2-x+(1)/(2)=0`

Therefore the coefficient of
`x^2` here is 1.

b) To make 5 the coefficient of
`x^2`

Simply multiply
`x^2-x+(1)/(2)=0` by 5

This gives:
`5x^2-5x+(5)/(2)=0`

c)
`-(1)/(2)`

To make
`-(1)/(2)` the coefficient of
`x^2`

Simply multiply
`x^2-x+(1)/(2)=0` by
`-(1)/(2)`

This gives:
`-(1)/(2)x^2+(1)/(2)x-(1)/(4)=0`

d) √3

To make
`√(3)` the coefficient of
`x^2`

Simply multiply
`x^2-x+(1)/(2)=0` by
`√(3)`

This gives:
`(√(3)) x^2-x√(3)+(√(3))/(2)=0`