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
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
To find the equation whose roots are alpha^2 and beta^2, we need to use the relationship between the roots of a quadratic equation and its coefficients.
For a quadratic equation in the form ax^2 + bx + c = 0, the sum of the roots is given by:
α + β = -b/a
The product of the roots is given by:
αβ = c/a
In this case, we are given that α and β are the roots of the equation 2x^2 - 7x + 4 = 0. So we have:
α + β = 7/2
αβ = 4/2 = 2
Now, we need to find the roots of the new equation. Let's call the roots of the new equation as α^2 and β^2.
According to the given information, we can write:
α + β = 7/2
α^2 + β^2 = ? (what we need to find)
We know that (α + β)^2 = α^2 + β^2 + 2αβ
Substituting the values of α + β and αβ, we have:
(7/2)^2 = α^2 + β^2 + 2(2)
49/4 = α^2 + β^2 + 4
α^2 + β^2 = 49/4 - 4
α^2 + β^2 = (49 - 16)/4
α^2 + β^2 = 33/4
So 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 + 2 = 0
Therefore, the equation whose roots are α^2 and β^2 is:
2x^2 - (33/4)x + 2 = 0