Question

If α and β are the roots of the equation

3x 2 +20x−7=0, find the equation whose roots are
α+ 1/β and β+ 1/α
a) 3x 2 −15x+7=0
b) 3x 2−26x+7=0
c) 3x 2 +26x−7=0
d) 3x 2 −15x−7=0

Answer 1

To find the equation with new roots, we use Viète's formulas to compute the sum and product of the original roots, α and β. We then find the sum and product of the roots α + 1/β and β + 1/α to construct the new quadratic equation.

Step-by-step explanation:

To find the equation with roots α + 1/β and β + 1/α, given the original equation 3x2 + 20x - 7 = 0 with roots α and β, we first use Viète's formulas to find the sums and products of α and β:

• α + β = -b/a
• αβ = c/a

Here, a = 3, b = 20, and c = -7, so:

• α + β = -20/3
• αβ = -7/3

We then find the sum and product of the new roots α + 1/β and β + 1/α:

• (α + 1/β) + (β + 1/α) = (α + β) + (β/α + α/β) = (α + β) + (1/(αβ))(αβ + αβ) = -20/3 + 2(-7/3)^(-1) = -20/3 + 6/7
• (α + 1/β)(β + 1/α) = αβ + α/α + β/β + 1/(αβ) = αβ + 1 + 1 + 1/(αβ) = -7/3 + 1 +1 + 3/(-7) = -7/3 + 2 - 3/7

The new quadratic equation will have the form x2 - Sx + P = 0, where S is the sum and P is the product of the new roots. We simply replace S and P with the values we calculated from the sum and product of α + 1/β and β + 1/α to get the new equation.