Home
Class 12
MATHS
If alpha is a root of the equation 4x^2+...

If `alpha` is a root of the equation `4x^2+2x-1=0,` then prove that `4alpha^3-3alpha` is the other root.

Text Solution

Verified by Experts

Let other root is `beta`,
then `alpha+beta=-2/4=-1/2` or `beta=-1/2-alpha`……..i
and so `4alpha^(2)+2alpha-1=0` because `alpha` is a root of
`4x^(2)+2x-1=0`
Now `beta=4alpha^(3)-3alpha=alpha(4alpha^(2)-3)`
`=alpha(1-2alpha-3) [ :' 4 alpha^(2)+2alpha-1=0]`
`=-2 alpha^(2)-2 alpha`
`=-1/2(4alpha^(2))-2alpha`
`=-1/2(1-2alpha)-2alpha [ :' 4 alpha^(2)+2alpha-1=0]`
`=-1/2-alpha=beta` [ from eq. (i) ]
Hence `4alpha^(3)-3alpha` is the other root.
Promotional Banner

Similar Questions

Explore conceptually related problems

If alpha,beta are roots of the equation 6x^(2) + 11x + 3 = 0 then :

If alpha and beta are roots of the equation x^(2) + x + 1 = 0, then alpha^(2) + beta^(2) is

If alpha is an imaginary root of the equation z^(n)-1=0 then 1+alpha+alpha^(2)+….. .+alpha^(n-1)=

If alpha, beta are the roots of the equation x^(2)-px+q=0 , prove that log_(e)(1+px+qx^(2))=(alpha+beta)x-(alpha^(2)+beta^(2))/(2)x^(2)+(alpha^(3)+beta^(3))/(3)x^(3)-...

If alpha, beta are the roots of the equation x^(2)+x+1=0 , then the equation whose roots are (alpha)/(beta) and (beta)/(alpha) is

If alpha,beta be the roots of the equation 3x^2+2x+1=0, then find value of ((1-alpha)/(1+alpha))^3+((1-beta)/(1+beta))^3

If alpha and beta are the root of the equation x^(2)-x+1=0 then alpha^(2009)+beta^(2009)=

If alpha, beta are the roots of the equation x^(2)-2x-a^(2)+1=0 and gamma, delta are the roots of the equation x^(2)-2(a+1)x+a(a-1)=0 such that alpha, beta epsilonn (gamma, delta) find the value of a .

If alpha and beta are the roots of the equation 2x^2-3x + 4=0 , then the equation whose roots are alpha^2 and beta^2, is

If alpha, beta be the roots of the equation x^2-px+q=0 then find the equation whose roots are q/(p-alpha) and q/(p-beta)