Final answer:
To find the quadratic equation whose roots are αγ+βδ and αδ+βγ, we use Vieta's formulas to determine the new sum and product of the roots, leading us to x² - 4x - 15 = 0.
Step-by-step explanation:
If α,β are the roots of the equation x²−3x+5=0 and γ,δ are the roots of the equation x²+5x−3=0, then to find the equation whose roots are αγ+βδ and αδ+βγ, we can use the relationship between the roots and coefficients of a quadratic equation. According to Vieta's formulas, for the equation ax²+bx+c=0, the sum of the roots is -b/a and the product is c/a.
For the first equation, α+β = 3 and αβ = 5. For the second equation, γ+δ = -5 and γδ = -3.
To find the new equation, we set up the sum and product of the desired roots: (αγ+βδ) + (αδ+βγ) and (αγ+βδ)(αδ+βγ). We use the fact that αγ+βδ = αδ+βγ = αβ + γδ (because α+β = γ+δ) and solve step by step.
Therefore, the sum of the new roots is = 2(αβ + γδ) = 2(5 + (-3)) = 4 and the product is (αβ·γδ) = 5·-3 = -15.
The required quadratic equation with these roots is x² - the sum of roots · x + product of roots = 0 which translates to x² - 4x - 15 = 0.