If alpha and beta are the root of the equation 2x^2-7x+4=0.Find the equation whose roots are alpha^2 and Beta^2

Answer 1

Let's use the Vieta's formulas to find the sum and product of the roots of the equation 2x^2 - 7x + 4 = 0.

The sum of the roots α + β is given by:
α + β = -(-7)/2 = 7/2

The product of the roots αβ is given by:
αβ = 4/2 = 2

Now, let's find the equations with roots α^2 and β^2.

The sum of the roots α^2 + β^2 is given by:
α^2 + β^2 = (α + β)^2 - 2αβ
= (7/2)^2 - 2(2)
= 49/4 - 4
= 49/4 - 16/4
= 33/4

The product of the roots α^2β^2 is given by:
α^2β^2 = (αβ)^2
= (2)^2
= 4

Therefore, using Vieta's formulas, the equation with roots α^2 and β^2 is:
x^2 - (α^2 + β^2)x + α^2β^2 = 0
x^2 - (33/4)x + 4 = 0