In the quadratic equation (k - 2)x^2 + 50x + 2k + 3 = 0, the roots are reciprocals of each other. Find the value of k.?

Consider this general form:

ax² + bx + c = 0

Let the roots be α and β. The roots have these two useful relations:

αβ = c/a
α + β = -b/a

In this case the roots are reciprocals.

αβ = 1
c/a = 1
(2k + 3)/(k - 2) = 1
2k + 3 = k - 2
k = -5

(k - 2).x² + 50x + (2k + 3) = 0

Polynomial like: ax² + bx + c, where:
a = (k - 2)
b = 50
c = (2k + 3)

Δ = b² - 4ac (discriminant)

x₁ = (- b + √Δ)/2a

x₂ = (- b - √Δ)/2a


The roots are reciprocals of one another → x₁ = 1/x₂

(- b + √Δ)/2a = 1/[(- b - √Δ)/2a]

[(- b + √Δ)/2a] * [(- b - √Δ)/2a] = 1

(- b + √Δ).(- b - √Δ)/4a² = 1

(- b + √Δ).(- b - √Δ) = 4a²

b² + b.√Δ - b.√Δ - Δ = 4a²

b² - Δ = 4a² → recall: Δ = b² - 4ac

b² - (b² - 4ac) = 4a²

b² - b² + 4ac = 4a²

4ac = 4a²

c = a → recall: c = 2k + 3

2k + 3 = a → recall: a = k - 2

2k + 3 = k - 2

→ k = - 5