If x₁ and x₂ are the roots of the equation x² + 2x − 15 = 0 then the quadratic…

Given quadratic equation:
x² + 2x − 15 = 0
Let its roots be x₁ and x₂.
For a quadratic equation ax² + bx + c = 0:
Sum of roots:
x₁ + x₂ = −b/a
= −2/1
= −2
Product of roots:
x₁x₂ = c/a
= −15/1
= −15
We need the quadratic equation whose roots are:
1/x₁ and 1/x₂
Sum of new roots:
(1/x₁) + (1/x₂)
= (x₁ + x₂)/(x₁x₂)
= (−2)/(−15)
= 2/15
Product of new roots:
(1/x₁)(1/x₂)
= 1/(x₁x₂)
= −1/15
Now form the quadratic equation:
x² − (sum of roots)x + (product of roots) = 0
x² − (2/15)x − 1/15 = 0
Multiply throughout by 15:
15x² − 2x − 1 = 0