Question

If α and β are the roots of the equation x^2 - 2x^2 - 5 = 0, find the equation whose roots are α^2β and αβ^2.

If α and β are the roots of the equation 3x^2 - 5x + 4 = 0, find the value of (α/β) + (β/α).
a) α^2β - 5αβ^2 = 0, 19/3
b) α^2β + 5αβ^2 = 0, -19/3
c) α^2β - 5αβ^2 = 0, -19/3
d) α^2β + 5αβ^2 = 0, 19/3

Answer 1

`c) α^2β - 5αβ^2 = 0, -19/3`

The equation with roots
`α^2β and αβ^2 for x^2 - 2x - 5 = 0 is (x^2 - 2x - 5)^2 - 2(x^2 - 2x - 5) - 5 = 0.` For the quadratic equation
`3x^2 - 5x + 4 = 0` with roots α and β, the value of
`(α/β) + (β/α) is -19/3.`

Step-by-step explanation:

The given quadratic equation is x^2 - 2x - 5 = 0. Let α and β be the roots of this equation. The product of the roots (αβ) is equal to the constant term divided by the coefficient of the quadratic term, which is -5/1 = -5. Now, we are asked to find the equation whose roots are α^2β and αβ^2. We know that if α and β are the roots of a quadratic equation, then the new equation with roots
`α^2 and β^2 is given by x^2 - (α + β)x + αβ = 0.` Therefore, the required equation is
`(x^2 - 2x - 5)^2 - 2(x^2 - 2x - 5) - 5 = 0.`

Now, let's simplify this equation:

`(x^2 - 2x - 5)^2 - 2(x^2 - 2x - 5) - 5 = x^4 - 4x^3 - x^2 + 8x + 20 - 2x^2 + 4x + 10 - 5 = x^4 - 4x^3 - 3x^2 + 12x + 25 - 5 = x^4 - 4x^3 - 3x^2 + 12x + 20 = 0.`

The roots of this new equation are α^2β and αβ^2. Now, for the second part of the question, we are given the quadratic equation
`3x^2 - 5x + 4 = 0 with roots α and β`. We are required to find the value of (α/β) + (β/α). Using the fact that α and β are the roots, we have
`α + β = 5/3 and αβ = 4/3. Now, (α/β) + (β/α) = (α^2 + β^2)/(αβ)`. Using the formulas
`α^2 + β^2 = (α + β)^2 - 2αβ,`we get
`(5/3)^2 - 2(4/3) = 25/9 - 8/3 = -19/3`. Therefore, the final answer is
`(α/β) + (β/α) = -19/3.`